BEVERLEY EDUCATIONAL SERIES 

EDITED BY 

W. W. CHARTERS 

PROFESSOR OF EDUCATION 
UNIVERSITY OF ILLINOIS 



SCHOOL STATISTICS AND 
PUBLICITY ^f; 



BY 

CARTER ALEXANDER 

FIRST ASSISTANT STATE SUPERINTENDENT OF PUBLIC 

INSTRUCTION FOR WISCONSIN 

SOMETIME PROFESSOR OF SCHOOL ADMINISTRATION 

GEORGE PEABODY COLLEGE FOR TEACHERS 




SILVER, BURDETT AND COMPANY 

BOSTON NEW YORK CHICAGO 



\r^ 



% 



%^^ 



•h^ 



Copyright, 1919, by 
CAETER ALEXANDER 

All rights reserved 



M":r 



n -; 



CI.A5L1824 



EDITOR'S PREFACE 

We hear from the forum and pulpit that reconstruction 
must follow a war of such magnitude as that just closing. 
But the laws of habit still operate and, if permitted to do 
so, the nation will return, in fact will prefer to return, to 
those accustomed grooves of thought and action from 
which they have been so vigorously shaken. 

This return is not possible in those cases which have an 
economic basis. The industrial world will be compelled 
by the insistence of labor to change its ideals and practices, 
and problems of finance, taxes, and revenues will force 
reconstruction in the field of politics. But education will 
not feel the insistent urge of economic forces unless some 
obvious catastrophe, such as the spectacle of schoolrooms 
unprovided with teachers because of public parsimony, 
visualizes the crisis for the taxpayers. Situations less 
obvious than this the'citizen does not see and understand. 
It is difficult to show him that low salaries may result in 
other evils as serious as the corporeal absence of teachers 
from the classroom. He must be shown that not only 
must teaching be good but that better buildings, better 
sanitation, curriculum changes, compulsory education, 
and vocational education must be provided. 

Sonorous or staccato generalities about reconstruction 
which, I venture to say, in less than a twelvemonth of 
our swift-moving current of national life will have be- 
come platitudinous, will not produce that sympathetic 
attitude toward educational reconstruction which is neces- 



vi Editor's Preface 



I 



sary for its processes to be carried on. Antebellum habits 
will wage a momentous warfare upon transitory emotional 
ideals. . 

And, even though the present moment should be preg-f 
nant with sympathy for the improvement of the schools, 
other obligations compete for a place in the necessarily 
limited field of public attention. National taxes com-i 
pete with school taxes, political conflicts are more promi-" 
nent than school necessities, and industrial readjustments 
obtain readier attention than educational reconstruction. 

To meet this situation the man of the hour is the super- 
intendent of schools. The problem of crystallizing the 
liquid desires of the public for educational progress is as 
squarely placed upon his shoulders as any human task 
has ever rested upon any individual. For education is 
personal. The nation is educated in community groups. 
Just as the squad is the unit of the army, the community 
is the unit of the nation, and just as the non-commissioned 
officer who leads the squad is the ''backbone of the army,'' 
so the superintendent who leads his community is the 
backbone of the forces of education. The educational 
army is just as dependent upon his intelligence and in- 
dustry as the fighting army is dependent upon the cor- 
poral. 

The chief weapon for leading the people of a community 
in educational activity is publicity. And this may be 
obtained in three ways. One method is to develop ex- 
cellent schools and let the work speak for itself through 
satisfied parents, loyal teachers, and efficient children. 
But this method as the sole method is open to the criti- 
cism that parents and children cannot clearly distinguish 
between good school systems and inferior systems, and 
to the further criticism that if people are too well satisfied 



Editor's Preface vii 

with their schools they are not sensitive to the need for 
better schools. So, important as the maintenance of a 
good system is, its presence does not insure knowledge of 
its excellence and needs. 

A second method is personal explanation of what the 
school is doing and attempting, carried on by conferences 
with the school board, by public meetings, and by private 
conversation. But a third method to be added to these 
is the superintendent's annual report and printed com- 
munications, in which a wider audience is reached. All 
three of these methods are used by those superintendents 
who have been most successful in molding public opinion. 

But, as Mr. Alexander points out, seventy per cent of 
a group containing one hundred twenty-eight members 
of two of the most distinguished organizations of an in- 
telligent community stated that they did not read the 
school board reports. And this proportion is small for 
the nation as a whole. 

Upon the problem of making the superintendent's 
report readable by his community this text is directed. 
The author attacks the whole problem from the collect- 
ing of the data and their statistical treatment, to the 
presentation of his findings in simple and graphic form. 
It is, therefore, presented as a notable attempt to make 
known to the public those inner workings of the school, 
to the end that fluid educational interest may take on 
stability of action directed toward progressive ends. 



AUTHOR'S PREFACE 

The admitted ineffectiveness of our school statistics is 
due mainly to two causes. First, many superintendents 
do not know how to apply statistics as well as do their 
equals in intelligence in other fields. Second, these super- 
intendents do not know — because they have been too 
busy to learn — how to present statistical matter to the 
public effectively. 

As the experience of publicists in other fields shows, the 
ways of doing these things are simple. But not enough 
advantage of the labors of these men has been taken, nor 
have their results been adapted for easy and quick use 
by the busy superintendent. There is pressing need for 
a book which will do these very things. The aim of this 
book, then, is to make available for superintendents and 
classes in school administration the results of years of 
study of statistical theory and its applications to school 
data for publicity purposes, as shown in school reports 
and surveys. 

These results have all passed through the fires of criti- 
cism of the large number of practical school men whom 
it has been the author's good fortune to have as students 
during the past six years. In particular, he wishes to 
acknowledge his great indebtedness to the men of Ed- 
ucation 245, his graduate course at Peabody, whose 
generous aid and criticism have contributed much to 
the practical helpfulness of the book. 

Acknowledgments are also gratefully made to the fol- 

ix 



X Author's Preface 



1 



lowing : Professor E. L. Thorndike of Teachers College, 
who introduced the writer to statistical method ; the 
editor, Professor W. W. Charters, who has given valuable 
suggestions for modifying the form of the book; Mr. 
W. C. Brinton and Professors J. F. Bobbitt and W. I. 
King, on whose writings the author has drawn freely; 
Dr. W. W. Theisen, supervisor of educational measure- 
ments in the Wisconsin State Department of Public In- 
struction, who has read part of the proof ; the various 
writers and foundations who have given permission for 
the reproduction of copyrighted charts; Messrs. C. H. 
Moore, E. McK. Highsmith, and S. C. Garrison for careful 
criticism of the manuscript. 

A special feature of this book is that all but five of the 
cuts are from drawings made by the boys in the Tech- 
nological High School at Atlanta under the direction of 
Mr. E. S. Maclin, at that time head of the drawing de- 
partment there. Mr. Maclin, who was one of the men 
in Education 245, has kindly given his services to produce 
this practical demonstration of the possibilities in utilizing 
high school students on school publicity. No doubt there 
are some slight errors or irregularities to be found in the 
cuts. But all such imperfections are only evidences of 
the genuineness of the demonstration. 

The school superintendent must be a publicist. He 
must make reports to the public. In many places for 
the next decade at least he must fight as hard as any officer 
in the trenches to ward off the incessant and fierce attacks 
made upon his school appropriations by politicians and 
hard-pressed but unthinking tax-payers. For warding 
off or beating back such attacks, his most effective weapons 
will be reports containing simple but skillful statistical 
devices for presenting the claims of the school children. 



Author's Preface xi 

Unless he has such weapons, the enemy will be liable to 
sweep over the schools and place them on a starvation 
diet. 

If this book in any material way helps the superin- 
tendent of schools, active or in training, to arouse his 
public and to secure adequate support for his school, it 

will fulfill its mission. 

Carter Alexander. 



SUGGESTIONS FOR USING THE BOOK 

I. TO THE SUPERINTENDENT IN THE FIELD 

The active superintendent, accustomed as he is to 
adapting materials quickly to his own problems, of course 
needs few "suggestions for using this book. But to save 
his time, a few are given. 

1. All of this book except Chapters V to VII can 
be read with little effort by any active school man. Ap- 
plications to his own work will constantly come to mind. 

2. A beginner in statistical method can attain a fair 
acquaintance with the matters involved in Chapters V 
to VII by a careful reading of the text. But a usable 
knowledge of these chapters, for one not previously familiar 
with statistical method, can hardly be expected unless the 
exercises or similar ones are done as indicated in the 
suggestions for the instructor of future superintendents. 
For such exercises, the superintendent generally has on 
hand several administrative problems that would be 
greatly simplified by adequate statistical treatment. 
His annual report, in particular, will furnish material for 
practically every exercise in the book. 

I 3. The table of contents and the index will enable him 
to use the book as a reference book for securing material 
or suggestions for procedure on school problems necessitat- 
ing statistical treatment. He can also quickly get ideas 
and devices for presenting his results to the public. 

4. If he has plenty of time, the procedure advocated 
for students in training will be very profitable. 

xiii 



xiv Suggestions for Using the Book 

II. TO THE INSTRUCTOR ENGAGED IN TRAINING FUTURE 

SUPERINTENDENTS 

The experience of the writer with the active superin- 
tendents and future superintendents in his own classes 
indicates the following : 

1. It is advisable to have a good supply of supple- 
mentary material for practice work and concrete illus- 
trations. At least one copy of each reference given Li 
the bibliography in the back of the book is advisable. 
In addition, one should secure before the class starts a 
fairly complete collection of typical annual reports and 
publicity pamphlets of the city and county superintend- 
ents in the district, state, and section of the country from 
which the students come. It is well to have several 
copies of the better ones of these, but for most of them 
one copy each is sufficient. These should be distributed 
among the members of the class, each being allowed to 
choose those he especially cares for and each being re- 
sponsible for his own set. All students may be required 
to become familiar with the better ones for which dupli- 
cate copies are available. 

2. Portions of the text may be prepared for class dis- 
cussion as in any textbook course. But the instructor 
may test the extent to which the students have mastered 
the text and create much more interest by constantly 
expecting them to bring in pertinent illustrations of the 
points under discussion. These illustrations should be 
obtained from the reports or other material for which 
the student is responsible, from his previous experience, 
or from current magazines and advertisements. 

3. A mastery of the text, however, can be assured only 
when students prove their ability to handle the exercises 



♦ 



Suggestions for Using the Book xv 

in this book, or similar ones which the instructor may 
easily make up or select from the references in the bibliog- 
raphy. ''To know whether any one has a given mental 
state, see if he can use it." 

Answers to exercises have been purposely omitted. 
Most students come to such exercises with a habit of 
striving for an accuracy to several decimals in the answer 
rather than of concentrating their attention on the prin- 
ciples involved. The particular decimals secured in any 
answer will often vary considerably according to the 
grouping used, the extent to which decimals are carried 
out, and so forth. Again, many administrative problems 
involve original data based on approximate measures, 
where attempts to secure absolute numerical accuracy 
are a sheer waste of time for everybody. By checking 
the procedure and answers of various students against 
each other, a sufficiently accurate answer will be ob- 
tained in the class. To secure this result, however, the 
instructor should see that the exercises are worked out 
in full and handed in. Judging by the writer's experi- 
ence, it is hardly worth while to discuss any of the ex- 
ercises that involve computation until they have been 
completely written out by the students. The work of 
both student and instructor will be greatly reduced for 
exercises which occur on pages 38 to 316, by utilizing the 
work already done on a problem, merely adding the work 
for the particular point to be illustrated and handing in 
the exercise again. 

If there is time and the students are sufficiently mature, 
the problem started in the exercises on pages 38, 43, 57, 
61, 81, 89 may be continued throughout the course by 
each student with excellent results. It is not continued 
in this book because of the difficulty of keeping a group 



xvi Suggestions for Using the Book 



together on such work in the usual time allotted to reci- 
tations. But in the writer's own classes, his students 
continue to work outside on such problems and make 
oral or written reports at the end of the course. Prob- 
lems of this nature interest students greatly. Any in- 
structor can find many opportunities for them in the 
local city school system, with resulting profit to the 
system. 

4. For the convenience of the instructor and a few of 
the students who evince unusual interest or capacity for 
the work, a few references are given at the end of various 
chapters. It is not, however, advisable to use many of 
these for class reading. In attempting to understand 
statistical method, the student must, above everything 
else, avoid confusion and keep his feeling of mastery over 
what he has done. But he will find the various writers 
taking up topics in different sequence and using different 
definitions or different technical terms in a manner very 
confusing to a beginner. Much better results will be 
secured if the instructor will use most of the references 
himself and adapt the additional material he wishes to 
the plan of this book and the special needs of his students. 

Experiments with the writer's own classes show that a 
satisfactory working knowledge of the material presented 
in this book can be comfortably acquired in from twenty- 
five to forty-five class periods. The shorter time will, of 
course, reduce the use of exercises. The longer time will 
allow abundant opportunity for the use of the exercises 
and special problems advocated. The book can thus be 
used in a six weeks' summer school, in a corresponding 
period during the regular session, or in whatever longer 
period is available. 



I 



TABLE OF CONTENTS 

PAGE 

Chapter One. Why We Need Better School Statistics 1 
I. Common Errors 
II. Wastefulness of These Errors 

III. Indifference of the PubHc 

IV. Progress Elsewhere in Using Statistics 

V. Unsatisfactory State of Affairs in School Statistics 

Chapter Two. Collection of Data 33 

I. When to Use Statistical Method 
II. How to Plan Statistical Treatment of Problems 

III. How to Determine Units and Scales 

IV. How to Do the Actual Collecting 

Chapter Three. Technical Methods Needed in School 

Statistics 90 

I. Usual Views 
II. Statistical Knowledge Needed for School Surveys 

III. Statistical Knowledge Needed for Reading Educa- 
tional Investigations 

IV. Illustration of Value of Statistical Method to the 
Superintendent 

V. Statistical Method as a Form of Expression 

Chapter Four. Scales, Distribution Tables, and Sur- 
faces OF Frequency . . . . . . 100 

I. Scales 
II. Distribution Tables 
III. Surfaces of Frequency 

Chapter Five. Measures of Type 124 

I. The Mode 
II. The Median 

III. The Average 

IV. Which Measure of Type to Use in a Given Dis- 

tribution 

xvii 



XVIU 



Table of Contents 



Chapter Six. Measures of Deviation or Dispersion 
I. Extreme Range Variation 
II. Quartile Deviation 

III. Other Percentile Deviations 

IV. Median Deviation 
V. Average Deviation 

VI. Standard Deviation 
VII. Deviations for Skew Distributions 
VIII. Which Measure of Deviation to Use in a Given 
Distribution 



PAGE 

149 



164 



Chapter Seven. Measures of Relationships . 
I. Relationships Inside of One Group 
II. Simple Relationships between Different Distributions 

III. Coefficient of Variability or Dispersion 

IV. Correlation 

Chapter Eight. Supplement on Statistical Treatment . 187 
I. Reliability of Statistical Results 
II. Special Economies in Calculation 
III. Combining Data Given in Rank Order Only 

Chapter Nine. Uselessness of Statistics in Current 

School Reports 200 

I. The Situation 
II. Causes of the Uselessness 
III. Devices for Effective Presentation 



Chapter Ten. Presenting Tabulated Statistics to the 

Public 

I. Possibilities of Using Tabulations of Statistics to In- 
fluence the Public 
II. How to Give a Bird's-Eye View through Tabulation 
III. How to Make Up a Series of Tables of the Same 
General Nature 

Chapter Eleven. Graphic Presentations of School 
Statistics Especially for the Public 
I. Object of Graphic Presentations 

II. How to Make Graphs for the Public from Statistical 
Data 



206 



234 



Table of Contents xix 



PAGE 



III. How Graphs for the Public Differ from Those for 

the Administrator 

IV. Examples of Good Graphs on School Statistics for 

the Public 
V. Economies in Making School Graphs for the Public 

Chapter Twelve. Translating Statistical Material 

ON Schools for the Public .... 303 
I. The Need of Translation 
II. Suggestions for Good Translations 
III. Examples of Good Translations of School Sta- 
tistics 

Selected and Annotated Bibliography .... 317 
Index 323 



SCHOOL STATISTICS AND PUBLICITY 

CHAPTER I 

WHY WE NEED BETTER SCHOOL S'^ATISTICS 

I. COMMON ERRORS 

Many school superintendents have no doubt laughed 
over some variation of the following : 

A clerk in a department store asked for a raise in salary and got 
this reply from the owner : 

"Why, this is no time to ask for more wages. Times are too hard 
and you do very little work. 

"I will show you how little work you do for me in a year. 

"The year has 365 days in it, and each day is 24 hours, divided 
in three equal parts, 8 hours for work, 8 hours for sleep, and 8 hours 
for play. 

"Now just listen. Take 8 hours that you sleep in each day, 
which is 122 days, from 365, and that leaves 243 days. 

"You play 8 hours each day, which is another 122 days from 243, 
and that leaves 121 days, — see? 

"Now, there are 52 Sundays when you do not work; just take 
these from 121, and that leaves 69 days. 

"When the summer comes, you say: *I can't work, I'm all in, 
and I want a vacation.' I give you 2 weeks off; take 14 days 
from 69. This leaves 55 days. 

"Then the store closes a half day every Saturday. This is 26 
from 55, which leaves 29 days. 

"Then you take li hours for your lunch each day, which is 28 
days, from 29, or 1 day left. 

"And I just remember that that day was the retailmen's picnic, 
and you asked off to go to it." 

1 



2 School Statistics and Publicity 

The isolated statements seem fair enough on the sur- l\ 
face when taken one at a time. What has been done 
to manipulate the figures so that a patently erroneous 
conclusion is reached ? Humorous as this is, it is not so 
far beyond what can be found sometimes in the statistical 1 
parts of school reports. 1 

Let us examine samples of these errors, which for con- 
venience may be regarded as arising from troubles with 

1. Unanalyzed Totals. 

2. Comparisons Employing Indefinite Units. 

3. Comparisons Using Unsound Elementary Treatment. 

4. Attempts at Too Great Accuracy. 

5. Neglect of Technical Statistics. 

6. Presentations of School Statistics to the Public. 

These will now be discussed in order, but only for the " 
purpose of showing the errors clearly. The remedies 
for such troubles will be given in subsequent chapters. ^ 

1. Unanalyzed Totals 

Some commercial clubs and school systems print figures 
on their letterheads giving the size of the schools, number 
of teachers, number of children enrolled, number of build- 
ings, size of plant, etc. A certain normal school in a 
middle-western state uses this material on its letterhead : 

A Teacher's College 

Faculty, 46 — men, 19 ; women, 27 

Enrollment, 1915-16, 2179 

Average attendance, 874 

1 For those who wish to use this book for reference purposes or 
for review, cross references to later portions are given throughout 
this chapter. But for those who intend to read the whole book and 
who are going through the chapter for the first time, it is advisable 
to ignore the cross references at this stage. 



Why We Need Better School Statistics 3 

These figures mean very little unless the reader knows in 
addition whether the proportion of the number of the 
faculty to the number in the student body is somewhere 
close to that of the average normal school in the country ; 
whether, in averaging the attendance, there was one 
short term with a heavy attendance set over against 
three long terms with a rather lean attendance, etc. 

Another illustration of this error is the statement 
sometimes made when medical inspection is installed 
in a city, that the results show 90 per cent or over of the 
children to be defective. This tends to alarm people 
until they consider that there must be some fallacy in 
it or there could be no such thing as normal children in 
that city. The fact is that practically every child is 
physically defective on some point. When children are 
measured on ten to twenty physical points, each child 
is certain to be defective on some one point, but many 
of them are, on the whole, average or normally healthy 
children. The unanalyzed total gives a misleading im- 
pression. 

2. Comparisons Employing Indefinite Units 

Even if the error of unanalyzed totals is avoided, a 
second weakness may creep in through foolish or mis- 
leading comparisons. These may be of this nature 
because of lack of suitable units for comparing the things, 
or the trouble may arise because of unsound methods 
in making the comparisons. 

Enrollment. For example, a certain college at one 
summer session announced an enrollment of 1108. At 
the second summer session after this the figures were 
1484. Apparently this was a gain of only 34 per cent. 
In reality the school had gained about 60 per cent. The 



School Statistics and Publicity 



^ 



enrollment the first time was padded by the issuing of 
visitors' cards, even to wives of the faculty. The enroll- 
ment the third time included only bona fide students, 
owing to the need of impressing certain people with the 
school's actual enrollment. The difference in units here 
prevented the school from gaining the advertising value 
of the really large increase of the second year. 

Serious errors often occur in comparisons between 
school systems because the number of children has not 
been figured on the same basis. Thus, this number may 
be calculated as the total enumerated of school age (school 
census), as the total ever on the roll (enrollment), as the 
total number in attendance that have not been absent 
over three days (number belonging), or as the average 
daily attendance. Also, the percentage of daily attendance 
is often figured on the " number belonging " base. This of 
course always gives a high percentage of daily attendance, 
since absent pupils are rapidly dropped from the count. 
It is in reality a measure of the holding power of the 
schools for truants, rather than for all children of school 
age. Sometimes the enrollment in a school system equals 
or exceeds the enumeration, not because of the attraction 
for enumerated children, but because of the enrollment 
of non-residents, no mention of which may be made in 
that connection. Or the enrollment may be greater be- 
cause of transfers inside the system so that the same 
pupil is enrolled in more than one ward school and con- 
sequently is counted twice or more on enrollment. 

School Expenditures. Erroneous conclusions are often 
reached in attempting to compare the total expenditures 
of one city school system with those of another city, 
without proper units. These totals are vitally affected 
by such factors as the number of people residing in the 



Why We Need Better School Statistics 5 

city, the per capita wealth, the proportion of children 
in the population, the rate of taxation for all purposes, 
etc. A rich suburban city can be expected to expend 
far more on its schools than can a city of the same size 
made up mainly of factory workers or similar wage 
earners. 

Distribution of School Moneys. In some places in 
the South, state school money is distributed on one unit, 
the enumeration basis, and expended on another, the 
number of children (more likely of white children) in at- 
tendance. For example, some years ago in Alabama one 
community is said to have had enough in this way to pay 
the principal of the white school $1200 and give him 
three assistants, all for an enrollment of forty-five white 
children and an average daily attendance of not more 
than thirty-five. This was of course achieved by spend- 
ing on the white schools considerable money drawn on 
negro census children. ^ 

Age of Child. Another indefinite unit is the " age " 
of a child in any given school system. Is a six-year-old 
child one that is six and has not reached seven, or is it a 
child from five and a half up to six and a half years old ? 
In making the recent survey of Cleveland, Dr. Ayres 
found that the age of a child in that city was determined 
by whatever birthday fell nearest to September of that 
particular school year. This meant that a child was the 
same age for a whole year. Children entering in June 
were set down as being the same age as they would have 
been if they had entered the preceding September. This 
made the average ages of the children somewhat younger 
than they 'should be. It had the effect of making many 
children that were over-age in the fall, no longer over-age 
1 Reported by Mr. W. F. Puckett 



6 School Statistics and Publicity 



1 



when promoted at the mid-term. Thus at the beginning 
of the second term there was always a large and instan- 
taneous drop in the percentage of retarded children, al- 
though the children in reality were relatively about as 
they were in September. That is, they were a half year 
older and a half year farther along in the grades. Be- 
cause of this peculiar definition of age, the survey com- 
mission could make no comparisons in acceleration, elim- 
ination, and retardation, with other cities.^ Similar 
difficulties confront a superintendent who figures retarda- 
tion before promotions in his system one year, and after 
promotions the next year. And when he compares his 
school system with any other system on retardation, he 
must be certain that he is taking it at the same time it 
was figured in the other systems. Another example is 
the custom reported from Louisville, Kentucky, of count- 
ing withdrawals as of the age they were on the previous 
September first. Many of those withdrawing after the 
middle of the year are, of course, in a higher grade or half- 
grade from what they were when their ages were taken. 
Teachers' Salaries. It is not uncommon to publish 
the salaries of teachers as being so much per month, with- 
out any mention of the number of monthly pajrments 
per year. Obviously the teacher has to live twelve months 
in the year, and for purposes of comparison the total for 
the year is the figure that must be used, especially if other 
people are to be influenced. For instance, a Kentucky 
superintendent in a small city reported that when he got 
his first month's salary check for $125, the cashier of the 
bank, who drew $90 a month, remarked jocularly that 
his own salary should be raised. He considered that the 
superintendent was earning more money. In fact, the 
1 Cleveland Survey, volume on "Child Accounting," pp. 40-43 



Why We Need Better School Statistics 7 

superintendent drew $125 a month for eight months or a 
total of $1000 ; the cashier drew $90 a month or a total 
of $1080. It is equally obvious that such a report on the 
monthly basis is very much to the advantage of the sys- 
tem that runs for a shorter term. 

Another example is the published salary list of the 
faculty in a certain southern normal school. Some men 
were listed at $150 and others at $133J a month. But 
as a matter of fact, the $150 men were paid for eleven 
months and the $133J men for twelve months a year, 
while all worked the same length of time, thus making 
an average difference of only a little over $4 a month. 

Teachers' Schedules. It is rather common to attempt 
to compare the amount of work done by teachers within 
the same high school, normal school, or college, mainly 
by taking simply the number of recitations each has per 
week. This assumes that one recitation means as much 
work as any other and that teachers do no work except 
that connected with recitations. It leaves out of account 
such factors as : one recitation with a large class means 
much more work than one with a small class ; two reci- 
tations in different classes involve much more labor than 
two sections of the same class ; teachers serving on cer- 
tain committees do a great deal of work connected with 
them and entirely apart from class duties. 

Tax Rate. Some state superintendents and many city 
superintendents have published comparisons of cities or 
counties with other cities or counties, based on the tax 
rate for school purposes. This is of no value unless one 
knows the rate of assessment and can thus arrive at the 
real rate of taxation. It is a well-known fact that there 
is an enormous variation in the rate of assessments in 
various cities; that no two are practically the same. 



8 School Statistics and Publicity ^^j 

Yet how often do we see in school reports long lists of 
figures giving the school tax levies of many cities chosen 
almost at random. But these do not impress the tax- 
payer. He knows whether his total school taxes are heavy 
or not and will pay little attention to figures that merely 
place his city low in the list of cities ranked according to 
rate of school tax.^ 

But even the actual rate of taxation is still of little 
value unless one knows how heavily the taxpayer is bur- 
dened with other taxes. For example, several years 
ago at Arkadelphia, Arkansas, the taxpayers protested 
against a school tax of seven mills (70 cents on the $100) ; 
they said that if this were added to their state school levy 
of three mills (30 cents on the $100), it would make their 
total school tax ten mills (100 cents on the $100), or higher 
than most places in the South. This was totally wrong, 
for many of the communities with which they were com- 
paring themselves paid a county school tax in addition to 
the local and state school taxes considered by them.- 

College Degrees. A similar trouble arises from the 
custom in many schools of at least secondary grade, of 
advertising the percentage of their faculty that hold de- 
grees. But with such degrees undefined these figures 
mean nothing. In all sincerity may not the public ask 
why an A.B. from Harvard or Vanderbilt University is 
rated just the same as one from a private college of junior 
rank which has perhaps four second-rate teachers and 
practically no equipment? The fact is that the public 
does ask just such a question. For this reason thinking 
people place very little value at the present time on the 
mere announcement of a degree. They wish to know 
what institution conferred it. As a consequence some 

1 Se3 pp. 51-2 2 Reported by Mr. A. E. Phillips 



I 



Why We Need Better School Statistics 9 

editors, authors, and catalogues, when giving a man's 
degrees, also indicate the institutions conferring them. 

Unequal Things. In the preceding examples the lack 
of suitable units has come about rather from thoughtless- 
ness than from any definite intention. But a very com- 
mon error in school statistics is caused by deliberately 
considering as equal, things which a very little serious 
thought would show are not equal. Professor Thorn- 
dike 1 gives a good example of the seriousness of this 
error. Dr. Rice had ranked a large list of words as of 
equal difficulty to spell. Professor Thorndike gave the 
same list to a group of children with these results : 42 
children missed necessary; 37, disappoint; and so on, down 
to 1, feather; none, picture. Making the most of the 
possibility that the group of children found some of the 
words easier than others because of recent drills, yet the 
fact is unmistakably clear that picture is not so difficult for 
these children to spell as is necessary. 

An especially good example of considering as equal 
things which are not at all equal is found in the 
following excerpt from a recent circular published in 
Virginia: ^ 

The general administration of school affairs in Virginia has also 
furnished striking evidence of sound economy as well as splendid 
efficiency. This is best shown, perhaps, in the recent comparison 
with the wealthy state of Minnesota, which has practically the same 
population as Virginia. In 1916 Virginia with eight millions of rev- 
enue enrolled 11,560 more pupils than Minnesota with twenty-six 
millions of revenue. 

This ignores the proportion of children in the two state 
populations, the length of school term, or anything con- 
nected with quality of education. The schooling given 
1 Thorndike, E. L. : Mental and Social Measurements, p. 8 



ft 



10 



School Statistics and Publicity 



the average child in Virginia certainly is nothing like 
the schooling given the average child in Minnesota. 
The latter is so much superior that it might con- 
ceivably cost over twice as much and still be really more 
economical. 

A third instance is found in library reports of various 
schools. These are often useless for comparative purposes] 
because some schools report all bulletins, agricultural I 
and census reports, and duplicate copies, as separate books 
and of equal value. 

A fourth example is that furnished by the vicious at- 
tacks on the state university made in some states by un- 
scrupulous politicians. In these the yearly cost of school- 
ing for a child in a rural school is compared with the several 
hundred dollars required to pay for a year's instruction 
of a university student. For this comparison one child 
in a rural school is deliberately considered to be equal to 
one student in the university. ^ 

Grading Standards. In this connection we must not 
overlook the errors that arise from differences in the 
grading standards of different teachers in the same school. 
Superintendents for purposes of records, determination of 
class honors, etc., generally assume that their teachers 
grade on something like the same standard. But, in 
fact, the almost countless articles and books on uniform 
grading published in the last few years demonstrate that 
all teachers if left to themselves will vary widely in their 
standards of marking pupils. 

In some schools elaborate statistics are kept to deter- 
mine the standing of pupils by marks, especially in high 
schools for class or graduating honors, etc. But these 
awards are really doubtful because no effort has been made 

1 See also p. 47 



Why We Need Better School Statistics 11 

to have the teachers mark uniformly. It is only when 
teachers mark on something like a uniform basis that the 
valedictorian may be easily found by averaging marks 
given by them. Without some system of uniform mark- 
ing, a student getting an average of 95 may really be of 
the same ability as another getting an average of 97. In 
the University of Missouri, before the adoption of its 
present system of uniform grading, it was well known 
that certain departments were '' snap " departments 
because of the high grades given in proportion to the 
amount of work required of the student. Professor 
Meyer in one of his earlier articles on uniform grading ^ 
shows that in the department of philosophy at this time 
55 per cent of all grades given were ''A," while in the 
department of chemistry only one per cent were '* A." 
Consequently, students of the university who wished to 
get into the Phi Beta Kappa fraternity tended to choose 
all their electives from the departments known to give 
many high marks. It may be taken for granted that few 
possessors of a Phi Beta Kappa key in this institution 
about this time had taken any more than was required 
in chemistry, but probably many of them had elected 
work rather freely in philosophy. The application of 
the same principle to the high school valedictorian needs 
no elaboration. 

Judging Contests. The same error is made in judging 
debating or essay contests with from, say three to five, 
judges, unless the judges are obliged to grade on the same 
scale. Suppose we have ten contestants, and three judges 
who grade in terms of their own judgment, the grades being 
as follows : 

1 Meyer, Max: "The Grading of Students," Science, 28: 246. 
(Aug. 21, 1908) 



12 



School Statistics and Publicity 





Marked by 


Marked by 


Marked by 


Average 


Contestant 


Judge A 


Judge B 


Judge C 


Mark 


1 


55 


82 


90 


75.6 


2 


95 


80 


94 


89.6 


3 


70 


82 


92 


81.3 


4 


85 


88 


94 


89.0 


5 


70 


82 


91 


81.0 


6 


84 


86 


96 


88.6 


7 


80 


86 


96 


87.3 


8 


75 


84 


96 


85.0 


9 


75 


80 


96 


83.6 


10 


60 


82 


92 


78.0 



Each of the judges has rated a different contestant as the 
highest, and the proposition is made that an average be 
taken of all the ratings. When this is done, it is found 
that Contestant 2 has won, although Judge B has rated 
him as one of the worst of the performers and Judge C 
has ranked him as tied for fifth place. Contestant 4 
loses although he is rated in first place by Judge B, second 
place by Judge A, and as tied for second place in Judge 
C's opinion. The reason he loses, although his combined 
rankings are higher, is this : the judges did not grade on 
the same scale. Judge A used a scale of much wider 
variation than did the other judges, and he gave Contest- 
ant 2 ten points more than any of the other contestants. 
The whole range of Judge B's grades was only eight points ; 
that of Judge C's only six points ; but Judge A's ranged 
forty points. Therefore, his rating of Contestant 2 more 
than overcame the combined judgments of the other two 
men.^ 

1 For further treatment of units and scales, see pp. 43-57. 



Why We Need Better School Statistics 13 

3. Comparisons Using Unsound Elementary Treatment 

Errors in the Use of Percentages. Of the unsound 
methods of comparison, some common ones deal with 
percentages. First, there is the kind that ignores the 
real meaning of percentage. For example, a superin- 
tendent may claim a decrease of 200 per cent in retarded 
children or number of withdrawals. Such a statement is 
of course meaningless, for there can be only 100 per cent 
of anything on which to figure the decrease. What is 
usually meant is that the present number is now only one 
third of what it once was, a decrease of 66| per cent. 

This error runs at once into the one which frankly loses 
sight of the original data. Some years ago a prominent 
city superintendent in Georgia had the charge brought 
against him that illiteracy of adult whites had increased 
100 per cent in his city, while in the state as a whole it 
had decreased from 13 per cent to 10 per cent or a 23 per 
cent actual decrease in the amount of illiteracy. (13 - 10 
= 3 . y3_ := 23 % + . ) The facts were that this city of about 
forty thousand population had, at the beginning of the 
period in question, a certain small number of adult white 
illiterates, say ten. During the period several families 
of ignorant Greeks moved in, bringing at least ten more 
adult white illiterates. Based on the original number 
of illiterates, the increase was 100 per cent; but based 
on the total population, the increase was negligible. Yet 
some supposedly well-informed people in this city, not 
knowing or thinking how few cases the misleading state- 
ment was based upon, were inclined to criticize the super- 
intendent of schools. 

Especially may the original data be forgotten in figuring 
percentages on parts of a small group. Thus, taking the 



k 



14 School Statistics and Publicity 



^1 



percentages of the various marks given by a teacher in a 
class of less than twenty pupils is of doubtful value. Any j 
unusual condition of any sort affecting one pupil only { 
will affect the number of pupils receiving the same mark, 
by more than one twentieth of the whole. If there are 
ten pupils in the class, it will be affected 10 per cent and 
so on. If two pupils out of ten received A, 20 per cent of 
the class would get this mark. If one of these pupils for 
any reason should do poorer work, the A group would 
lose 10 per cent of the whole, but half, or 50 per cent, of 
itself. 

In similar fashion the Missouri School Journal some 
years ago ran a comparison of state teachers' associations, 
based upon the percentage of teachers in the state attend- 
ing them. Rhode Island came first and Tennessee last. 
This frankly ignored to a considerable extent the original 
data with its lack of a unit for " teacher in attendance." 
The Rhode Island teachers could attend their meeting 
at Providence on a nickel street car fare or a dime at the 
highest. But many teachers in the larger states could not 
attend their state association for financial reasons. Thus 
a low percentage of attendance in some states might mean 
a much greater spirit and devotion than would a 100 per 
cent attendance in Rhode Island. 

Other comparisons attempt to relate directly groups of 
data which should first have their component parts turned 
into percentages. For example, on the wall in a certain 
state department of education, the grades on certificate 
examinations made in each subject are plotted in a graph 
ranging from 70 to 100. The name of the subject is put 
on the graph and the name of the person grading the 
papers. The attempt is made to standardize the gi'ading 
in the different subjects by comparing these graphs. But 



• 



Why We Need Better School Statistics 15 

because of the failure to turn the numbers of grades for 
each group into percentages of the whole, the compari- 
sons are either very difficult or erroneous. The compari- 
sons could be made directly only if there were the same 
number of papers in each subject, a thing which does not 
often happen even approximately. 

Carelessness in Securing Data. A second form of un- 
sound comparisons frequently comes from carelessness in 
securing data. For example, in comparing school sys- 
tems on expenditures, the figures taken for one year may 
be wholly misleading for at least one or two cities out of 
any twenty. An unusual condition in some one city, such 
as an epidemic or a great expansion of work, or the erec- 
tion of a building a few years before, may make the ex- 
penditures for that year wholly different from the usual 
expenditures of that city. Again, unless great care is 
taken, the total expenditures for one city may include 
money spent on buildings and repairs, and duplicates, 
while for other cities they cover current expenses only 
without duplicates. Unless the figures for each city are 
usual, or " average " ones, and unless they are taken for 
all cities on the same basis, no amount of care later will 
give sound comparisons. 

Omission of Important Factors. A particularly un- 
fortunate form of comparison is that which presents data 
related to each other and then draws from them conclu- 
sions that are entirely erroneous because certain impor- 
tant factors have been left out of consideration. In other 
words, it is the old fallacy of arguing from insufficient 
data. For instance, take the case of the state superin- 
tendent in the South who some years ago claimed credit 
for the increase in school revenues in his state. Both 
from the platform and in press reports, he claimed that he 



16 School Statistics and Publicity 

had done more towards adding money to the state school 
fund than had any of his predecessors. He even issued 
a bulletin in which he compared the amount per capita 
for schools during his regime with that of five of his prede- 
cessors in the office. In his closing sentence he styled 
himself " the wizard of finance." As a matter of fact, 
the state in question through industrial development had 
largely increased its wealth so that the same rates for 
school taxation brought larger school revenues. The 
wealth had increased faster than the population, and so 
the per capita spent on schools rose. However capable 
a state superintendent of schools may be, he can hardly 
legitimately claim much credit for increasing the wealth 
of his state in a few years, especially at a rate faster than 
the population. 

Another example is the argument sometimes used 
against compulsory education. This cites that illiteracy 
has decreased faster in certain southern states that have 
no compulsory education laws than in certain northern 
states that have had such laws for years. But it leaves 
out of account the fact that the smaller northern decrease 
in illiteracy is due to recent immigration of foreign illit- 
erates, a class of which only a small number come to the 
South. 

4. Attempts at Too Great Accuracy 

Cost Figures. Occasionally time and effort are wasted 
by the superintendent's going to great extremes to show 
accuracy in his figures. This is particularly unfortunate 
at times, because what is considered great accuracy may 
be only unnecessary work that can never make things 
accurate. For instance, a superintendent may go to 
great length to calculate the cost of instruction per day 



Why We Need Better School Statistics 17 

per child in his system, for comparison with similar figures 
for other systems. He runs his figures out to hundredths 
of a cent. But owing to the trouble in units previously 
mentioned, his results are really worth very little. One 
system has taken the school census for the number of 
children; another, the enrollment; and still another, 
the average daily attendance. Unless he knows that the 
same base has been taken for each system, no amount of 
mechanical accuracy later will give exact results. 

In many public presentations of school statistics, much 
of the effect is lost by attempting to be too accurate in 
giving the figures to the cent or the fraction thereof in all 
places. The mind of the average citizen, or even of an 
expert for that matter, cannot take in too many details. 
The exact figures add nothing to the impression in his 
mind, and indeed detract from the main things. As an 
illustration of this point, take Professor Bobbitt's expres- 
sions in the San Antonio Survey that " English costs in 
the neighborhood of $210,000 " and " spelling costs in 
the neighborhood of $40,000." ^ These are far more ef- 
fective than if he had run the sums out to the exact num- 
ber of dollars and cents obtained from a very long and 
tedious computation. 

Standards. A superintendent may make an unwise 
recommendation when he advocates that his school 
should reach the precise figure on an '' average " or 
" middle figure " of a certain table, comparing his school 
with other systems. In view of the well-known chances 
for inaccuracy in the original figures, it is far better to 
take for granted that such hair-splitting accuracy is not 
desirable and to use a rather wide standard. Professor 

1 Bobbitt, J. F. : Survey of San Antonio Public Schools, pp. 99, 
103 



18 



School Statistics and Publicity 



Bobbitt 1 does this in his " zone of safety '^ standard 
which includes the middle half of the group. As an il- 
lustration of the workings of his standard, we quote 
Table I on the costs of instruction in high school mathe- 
matics per one thousand student hours, in certain cities 
of the country. 

Table 1. Bobbitt Table Showing Cost of Instruction per 
1000 Student Hours (Mathematics) 

Cost per 
Name of school 1000 stu- 

dent hours 

University High . . , . $169 

Mishawaka, Ind , . . . . 112 

Elgin, 111 100 

Maple Lake, Minn » . . . 100 

Granite City, 111 88 

East Chicago, Ind 82 

De Kalb, 111. . T~~ 74 

San Antonio, Tex 69 

Harvey, 111 69 

Waukegan, 111 63 

South Bend, Ind 62 

East Aurora, 111 61 

Rockford, 111 59 

Booneville, Mo 58 

Brazil, Ind 56 

Leavenworth, Kan 56 

Greensburg, Ind 54 

Morgan Park, 111 53 

Noblesville, Ind 52 

Norfolk, Neb. \ . ~. ~. ! '. ] '. \ ] ] ] ] ] ] 42 

Washington, Mo 41 

Bonner Springs, Kan 38 

Russell, Kan 34 

Junction City, Kan 33 

Mt. Carroll, 111 30 

1 Bobbitt, J. F. : "High School Costs," School Review, 23 : 505-534. 
(Oct., 1915) 



Why We Need Better School Statistics 19 

From this table, Professor Bobbitt would not say that 
the cost of teaching high school mathematics should be 
exactly $59; but that it should be somewhere between 
$52 and $74. ^ This measure must for another reason 
be used with still more caution. In this same study by 
the same method, Professor Bobbitt found the '' zone of 
safety " for Latin to be from $54 to $92, while that for 
English ranged only from $43 to $67. That is, the zones 
differ for different subjects, and one cannot at first glance 
be sure that one ought to strive for a difference in costs 
in. the subjects. Again, this measure is not good for 
publicity purposes unless the superintendent's system is 
below the '' safety zone." In such cases it is very effec- 
tive. But if his system is above this zone, citizens are apt 
to rest contented or even to contemplate reducing school 

taxes. 

5. Neglect of Technical Statistics 

Averages. Even when the data are accurate, errors 
may creep in because of neglect of the elements of tech- 
nical statistics. A frequent example of this arises from a 
very loose use of the average to typify a group, especially 
in regard to teachers' salaries. In a recent investigation 2 
of salaries paid by fifteen of the best-known colleges for 
women, it was found that the average lowest salary paid 
was $700 and the average highest was $1500. The 
average would give a very erroneous impression, as far 
more of the instructors received the lower salaries than 
the higher. The maximum amount reported was $3000 ; 
the lowest, $100 and home. It is obvious that the high 
salary would exert a greater influence on the average than 
the lower one, since it is twice as far away. 

1 To be still more accurate, between $47 and $78. See p. 130. 

2 Journal of Pedagogy, 19 : 185 



20 School Statistics and Publicity ^^B 

This point may be further illustrated from a report of 
a state superintendent. ^ In one table he gives the highest 
and the lowest salary paid per month to male and to 
female teachers, for both races, separately. The table I 
following, in the same way, gives the average salary of 
the various groups. For example. County A reports 
the highest salary for white male teachers, $175 per month ; 
the lowest, $40; the average, $73.07. The fact un- 
doubtedly is that in County A more white male teachers 
get less than $73.07 than get more. This table is made 
worse by averaging the salaries paid both white teachers 
and colored. This average means little for two reasons : 
First, there are fewer colored teachers by far in Florida 
than white ; second, the colored teachers do not get any- 
thing like so much salary as the white. The result is 
that the average is higher than all or most of the salaries 
paid colored teachers, and lower than all or most of the 
salaries paid white teachers, and so is significant for 
neither group. 

In the summaries of another report of the same super- 
intendent of the same state for 1913-14, this error appears. ^ 
The average salary per month paid teachers in the ten 
highest and the ten lowest states in the United States is 
given. Wisconsin stands at the top of the lowest ten. 
Florida does not appear, so we suppose that she is some- 
where between the highest ten states and the lowest ten ; 
hence the apparent conclusion that Florida pays a better 
monthly salary than does Wisconsin. But from another 
table we learn that the average school term in Wisconsin 
is 175.7 days ; for Florida, 122.2 days. In other words, 
Wisconsin pays on the average for 53.5 more days than 

1 Report of State Superintendent of Florida, 1912 : 471-472 

2 Report of State Superintendent of Florida, 1913-14 : 37 



Why We Need Better School Statistics 21 

does Florida. Now if these averages had been expressed 
in years instead of months, the chances are very likely 
that Wisconsin would be higher up in the list of states 
than Florida. 

A very inaccurate but common way of taking the aver- 
age is illustrated by the following procedure which came 
under the writer's notice : In a study for school purposes 
of the negroes of Texas, the percentage of the negro pop- 
ulation to the total population of the state was desired. 
Three counties with a high percentage, three with a mod- 
erate percentage, and three with a low percentage were 
taken for the whole state. Then the average of these nine 
counties was determined. This is a very rough and in- 
accurate method. If the number of counties in the state 
were divided into groups having the same ratio to each 
other as the counties taken in the study, and if these latter 
were then chosen at random or equally spaced in their 
groups, the results would be fairly accurate. But such a 
state of affairs could hardly be hoped for in popular 
sampling of this kind. 

A similar error is liable to occur in the common practice 
of judging the work of a class by examining one of the 
best notebooks, one of the medium group, and one of 
the worst. Unless care is taken, the extremes may re- 
ceive an undue amount of emphasis. The medium group 
is always so much larger than the extreme groups that it 
ought to have at least two samples to each one for the 
other groups. 

Variations. Often bare averages are given, when varia- 
tions from the average are the significant thing. Suppose 
that two boys in school have the same average for their 
grades in different subjects, say 85. But one boy made 
these grades, 84, 86, 86, 88, 81, and the other made 97, 98, 



22 School Statistics and Publicity 

90, 80, 60. The deviations from the average in these two 
instances show that one boy is an '' average " boy in all 
his studies while the other is very fine in some, possibly 
those he likes, but poor in the others. 

In the illustration of the salaries paid teachers in col- 
leges for women, previously given, the significant thing 
is not the average but the deviation from that average. 

Professor Bobbitt in his San Antonio survey has one 
rather incomplete treatment in this matter of deviation, 
occurring in what is otherwise a most admirable use of 
variations.^ He desires to make the point that the 
problem of heating the schoolrooms in the city is of minor 
importance and gives a table of mean hourly temperature 
of the city for all school hours, month by month. This 
of course allows for deviations. These figures are for the 
winter months not a tremendous distance below the 68 
degree standard for comfort. This would indicate that 
the city was always mild in winter. However, the sig- 
nificant thing is not this mean temperature of San An- 
tonio, but the deviations from it. To get a mean temper- 
ature of say 55, there must be days below it, possibly a 
considerable number. If there are only a few days each 
month, it is clear that the school buildings must be 
equipped to keep children warm on those days or else 
school must be dismissed until it gets warmer. 

Sampling. An extremely common error in school 
statistics arises because of ignorance of " sampling." 
This corresponds to the error in logic arising from con- 
clusions drawn from one case, from too few cases, or from 
cases not properly selected. In many school statistics 
the samples are too few, or they are not impartially taken. 
Especially is this true in most questionnaire or straw 
1 San Antonio Survey, pp. 226 j^. 



Why We Need Better School Statistics 23 

ballot investigations. A questionnaire is generally an- 
swered chiefly by those especially interested in the subject. 
Any conclusions from such data should not be interpreted 
as applying to any save that class. The methods of taking 
the average described on page 21 are of course illustra- 
tions of bad sampling. 

Incorrect sampling appears often in school advertising. 
Thus, a leading southern university gives prominence in 
its advertising to the number of its graduates who have 
been members of Congress, governors of states, United 
States senators, or even President. In many ways of 
course such advertising is thoroughly justifiable, but not 
when it seeks to convey the impression that the typical 
student at the university will later reach such prominence. 
Again, a certain southern university appears to admit to 
graduate standing all graduates of a certain normal 
school which does about two years of college work. This 
graduate standing is granted because at one time three or 
four students from the normal school in question were 
taken on trial at this university and did good graduate 
work. That is, all students from the normal school are 
assumed to be as capable as the original three or four. 
Practically every school system, especially a high school 
striving for recognition, cites the performance of its best 
students in creating a general impression of its work. 
But such sampling needs to be taken with considerable 
salt. We might with equal justice expect all Chinamen 
to be so many Confuciuses, or all Americans to be so many 
Woodrow Wilsons. 

The process of sampling is treated at length later on.^ 
But briefly, where sampling is resorted to, it must be done 
purely at random. And enough cases must be taken to 

1 Pp. 62-71 



24 School Statistics and Publicity 

insure an approximately accurate impression of the entire 
distribution that is being sampled. 

Number on a Scale. A final technical weakness, one 
requiring great care to avoid, comes from a wrong inter- 
pretation of what a number means on a scale. For ex- 
ample, a score of 6.25 on any test should mean one-fourth 
of a step above the "6" step's starting point. If this 
step runs from 6 to 7, obviously the score of 6.25 is just 
what we need. If '' 6 '' is regarded as running from 5.5 
to 6.5, then 6.25 may really mean 5.75. A superin- 
tendent may measure his work on a scale by one method 
and compare it with work measured by another method 
on the same scale, and feel much elated over beating the 
other man by half a step, when as a matter of fact the 
half step advantage is due solely to differences in counting 
on the scale. 1 

6. Presentations of School Statistics to the Public 

When it comes to presenting school statistics to the 
public, the errors are even greater. There is often little 
or no tabulation, or there are such complications that 
they can be understood only by experts. Many of the 
presentations make no use of graphs, and others contain 
such intricate graphs or charts that laymen cannot easily 
understand them. Some of the charts have the zero line 
cut off so that the effect on the reader is apt to be totally 
misleading.2 In other graphs the scales used may give a 
false impression. 2 

II. WASTEFULNESS OF THESE ERRORS 

The mere errors of school statistics are not their worst 
feature ; it is the amount of time and energy put into such 

1 See p. 49 2 See p. 283 ^ g^e p. 284 



Why We Need Better School Statistics 25 

useless things. Some school executives apparently de- 
vote much time and effort to the collection and publica- 
tion of educational statistics of this useless sort, pre- 
sumably with the idea of doing some good. Professor 
Hanus ^ in a recent study of superintendents' reports 
found this to be true of one city : The report of the super- 
intendent contained 35 tables, 23 of which contained 
statistics for the year 1913-14; the other 12 presented 
comparative statistical summaries covering the period 
from 1906 to 1914 ; but only two of the latter group of 
tables had any relation to the first-named group. Forty- 
two per cent of all the reports studied by Hanus con- 
tained *' few or no comparative statistics, and very 
little or no satisfactory interpretation of statistics." 
The significance of such statistics is practically nil, he 
concluded. 

Snedden and Allen, in a book published sorhe years ago,^ 
indicate that in their investigation of the school reports 
of the best city systems of the country, much useless 
statistical material was found . This material they roughly 
classified as of two kinds : First, material that would be 
useful if properly related and explained ; second, material 
useless in itself, either because it has no significance what- 
ever, or because it is in detailed form, when only the sum- 
mary is the significant thing. 

The protests against such ineffectiveness in school 
statistics arise also from other sources. Superintendent 
Maxwell of New York frankly says : " There are some 
ways in which the efficiency of a school may be deter- 
mined with an approach to accuracy and without the 

1 Hanus, Paul : "Town and Gity Reports," etc., School and Society, 
3 : 145-155, 186-198. (Jan. 29, Feb. 5, 1916) 

2 School Reports and School Efficiency, pp. 1-10 



26 School Statistics and Publicity 

assistance and without the retardation of time-wasting, 
energy-destroying statistical research. 

'' There may be ways in which the so-called scientific 
surveys or investigations, when stripped of past and 
present absurdities, will help in determining efficiency." ^ 

Again, the writer of a very thought-provoking satire 
on present-day educational theory offers the following 
'' statistics on the different classes of teachers with re- 
spect to ' pedaguese ' or the scientific terminology em- 
ployed by educational theorists " 2 ; 

Use and think they understand it 12 % 

Have used and thought they understood it, but don't now . 2% 

Think they understand it, but don't use it 6% 

Use it but don't understand it 9% 

Don't use it, don't understand it, but esteem with awe those 

who do 51% 

Think it is rot 20% 

100% 
III. INDIFFERENCE OF THE PUBLIC 

Moreover, the public, particularly the influential 
portion of the public, have a distrust of these school 
statistics and are little affected by them. It may be that 
the average citizen has a wholesome respect akin to awe 
for any statistical presentation, as claimed by some 
writer in the Unpopular Review.^ In this article appear 
some very illuminating examples of misleading statistics 
in general fields. Errors similar to these can be found 
in many school reports. But the writer's experience 
and results from conferring with many practical school 
men point in the other direction. Many influential 

1 Maxwell, W. H. : "How to' Determine the Efficiency of a School 
System," American School Board Journal, May, 1915 

2 Henrick, Welland : A Joysome History of Education, p, 54 
8 May- June, 1915:352-366 



Why We Need Better School Statistics 27 

people who are not especially trained in statistics, are 
frankly skeptical of any statistical report on schools. At 
the same time they may be vitally interested in the schools 
and have confidence in the superintendent. 

School statistics are usually included in large reports 
which do not reach the public to any appreciable extent. 
Professor Hanus sent 250 letters to the members of the 
Harvard Club, and 250 to the members of the Chamber 
of Commerce in Boston. He asked these people if they 
had seen a report of the school board, including that of 
the city superintendent, within the last two years. He 
received 128 replies from members of the club and 83 
from members of the Chamber of Commerce. About 
70 per cent of these answered ''No." Snedden and Allen 
assert that the reports of city systems are not read and 
give as the reason that they are not so arranged as to be 
intelligible to the ordinary citizen. The common practice 
of issuing summaries or abstracts of school reports or 
surveys, indicates the same thing. 

The indifference of the public is further shown by the 
fact that editions of school reports are usually very small. 
Only a few years ago the proposition was made to place 
copies of the New York City report in the hands of each 
of the 15,000 teachers, so that more intelligent and ear- 
nest work would result. It was objected to on the ground 
that it would cost too much. But the cost of the reprints 
would have been insignificant compared with the $36,000,- 
000 spent on the public schools that year. Another 
illustration comes from Mobile. This city has a debt of 
$4,000,000, and an annual expenditure of about $464,000 
with $110,000 spent on schools. But the board hesi- 
tated about paying the $500 necessary to print the recent 
school survey. 



28 School Statistics and Publicity 

IV. PROGRESS ELSEWHERE IN USING STATISTICS 

While these and similar errors continue in statistics 
of many school systems with resulting indifference of the 
public, remarkable improvements and extensions in the 
use of statistical method have been made in other fields 
and in some schools. Whether we look in economics, 
sociology, census reports, insurance, cost accounting, 
biological sciences, or the best school reports, wherever 
scientific method is used, that is, practically, wherever the 
experimental method is used, there has been a refinement 
of statistical method and an application of it to the prob- 
lems in question. 

Let us list at random some of these problems as they 
come to mind : 

Life insurance calculations, by which the average number of years 
a man will live after any given age, is known. 

Weather conditions (rain, temperature, etc.) over any given area 
for a long period of time, by which it may be known whether it will 
be profitable to grow certain crops in that region. 

The adaptation of the average to measuring and sampling processes, 
especially of grain. 

The movement from country to city and its results. 

The tenancy problem, its causes, types, and effects on general 
social status of the region under consideration. 

Relative attainments of city and country born children. 

Relative strength of various European belligerents. 

The relation of men and women in various mental processes. 

The effect of factory labor on children. 

The fight against preventable diseases. 

Comparison of the intellect of the white man with that of the negro. 

The influence of woman suffrage in national elections. 

Within the field of educational investigation itself, it 
is significant to scan the list of publications at a graduate 
school of education like Teachers College (Columbia 
University) and see that from only a small number of 



Why We Need Better School Statistics 29 

dissertations embodying a few statistics simply handled, 
there has been a steady drift toward investigations in- 
volving the comprehensive use of statistical method of 
greater and greater accuracy. The School of Education 
at the University of Chicago requires a course in statis- 
tical method very early in the program of every graduate 
student. 

Statistical method has been used to get at the truth of 
many educational problems, ones that could not other- 
wise be solved, for example : 

Do women teachers drive boys out of high schools? 

Is the South taxing itself as high for schools as the North? 

Is the record on college entrance examinations as good an index 
of the student's later achievement in college as is his high school 
record or the college record of his brother? 

What is the relation between reasoning and fundamental opera- 
tions in arithmetic? 

What are the most economical ways of memorizing? 

Do teachers get more salary for each year of training beyond the 
high school? 

What are the causes of elimination of children from school? 

A similar progress has been made in these other fields 
in presenting statistical information to the public. Po- 
litical parties, labor leaders, Y. M. C. A. leaders, phil- 
anthropic workers, various foundations, bureaus of mu- 
nicipal research, evangelists, advertising agencies, and 
corporations are showing in their reports great progress in 
pictorial and graphic ways of presenting statistical mate- 
rial so as to influence the layman. 

V. UNSATISFACTORY STATE OF AFFAIRS IN SCHOOL 

STATISTICS 

In view of the progress in the use of statistics in other 
fields and in special educational investigations, our present 
unsatisfactory state of general school statistics cannot be 



30 School Statistics and Publicity 

allowed to continue. Statistical methods apply wherever 
things are to be counted or measured. Nearly all the 
problems of the school executive involve numerical data 
and cannot be adequately handled without statistical 
method. For example, what problem would a super- 
intendent have that did not relate to one of the following 
general fields? 

1. Children to be educated or changed. 

2. The aims of education, or the nature and amount of change to 
be produced in these children. 

3. The agents of this education — teachers and others. 

4. The means of this education — buildings, books, laboratories, 
etc. 

5. The methods by which these agents use these means. 

6. The changes resulting from various combinations of these with 
the first, which is the big thing in education .^ 

Every one of these fields affords numerical data for the 
solution of problems, and they cannot be solved without 
handling such data. In such fields '' the number of use- 
ful studies to be made is, for all practical purposes, in- 
finite." 2 

The school executive has as great need to appeal to the 
layman with statistical matter on schools, as do any of 
the publicists in other fields. In particular, he has prac- 
tically to make the same appeal for funds as do workers 
in these other fields, especially the religious and philan- 
thropic ones. Consecjuently, the school superintendent 
needs for school purposes any valuable methods or dis- 
coveries on statistics worked out in other fields. 



1 Thorndike, E. L. : "Quantitative Investigations in Education," 
School Revietv Monographs for College Teachers of Education, No. I, 
pp. 31-32 

2 Ibidem, p. 34 



I 



Why We Need Better School Statistics 31 

In the preceding, the aim has been to set forth clearly 
typical errors in school statistics and to indicate briefly 
the causes for such ineffectiveness. The rest of this book 
shows in detail how to work up school statistics properly 
and how to present them effectively to the public. 

Chapter Two deals with the collection of data. Chap- 
ter Three discusses the need for technical methods in han- 
dling school statistics. Chapters Four, Five, Six, and 
Seven show how to apply these technical methods to school 
facts that are to be presented to the public, — Chapter 
Four dealing with scales, distribution tables, and surfaces 
of frequency; Chapter Five with the three measures of 
type ; Chapter Six with the measures of deviation or dis- 
persion; Chapter Seven with measures of relationship. 
Chapter Eight is a supplement on statistical treatment, 
dealing with additional problems which the superintendent 
will encounter in working up his school statistics to present 
to the public. Chapter Nine treats of the ineffectiveness 
with the public of the statistics in many school reports. 
Chapter Ten shows how to tabulate school statistics for 
the layman. Chapter Eleven deals with graphic presenta- 
tions of school statistics for the public. Chapter Twelve 
gives effective ways of translating school statistics into 
words, for popular consumption. 

EXERCISES 

1. When may a child be counted as a six-year-old? 

2. Just what is the justice of a state university's rating a high 
school on the performance of its former graduates in the first two 
years at the university? 

3. Precisely what does a boast in the state teachers' journal that 
a certain girl graduate in a given high school made an average of 98.5 
for the four years, mean to other high school teachers? 

4. How valuable is the practice of some teachers of keeping 



32 School Statistics and Publicity 

records of the work of pupils for only the last few days before the end 
of the term ? 

5. How significant is the claim of a superintendent that his high 
school is better than that of a neighboring superintendent because 
he has in it 6 teachers as against the other's 4, and enrolls 60 non- 
residents as against the other's 40? 

6. Of what worth is the statement of a superintendent that he 
has increased the size of the graduating class in his high school 75 
per cent in two years? 

7. How could you compare in the same high school the cost of 
teaching one period a day for the principal who teaches 4 periods a 
day, with that for the Ej;)glish teacher who has 6 periods a day; with 
that for a science teacher who has 4 classes with double periods each, 
a day? 

8. What examples of the errors given on pages 2-24 have you come 
across in your previous experience with school statistics, and in just 
what did the trouble consist? 

9. Take some school report or some article in your state teachers' 
journal that contains considerable statistical material. Locate the 
errors, if any, in it and show in just what they consist. 



CHAPTER II 

COLLECTION OF DATA 

I. WHEN TO USE STATISTICAL METHOD 

1. When Statistical Method Is Profitable 

Which school problems may be subjected with profit 
to statistical treatment? The superintendent needs to 
know this, for we have seen how easy it is for him to 
waste his time and make errors in school statistics. 
These troubles frequently come from two ever-present 
dangers: starting on the details of the task before it 
has been clearly thought out; and attempting to use 
statistical method where it is not applicable or will not 
yield results worth the effort. In other words, two 
questions at once arise : 

1. Which problems of the superintendent may profitably be sub- 
jected to statistical treatment? ' 

2. How may he know whether a given problem can profitably be 
so treated? 

In reply to the first of these questions, it may be said 
that practically all of the superintendent's problems may 
be statistically treated, because on all of them he can 
work with data that may be counted or measured more 
or less accurately. The quotation from Professor Thorn- 
dike, page 30, shows how wide a range of problems can 
be thus treated. But to be still more specific, the follow- 
ing list of problems ever present for the superintendent 
must be so treated : ^ 

1 Made up chiefly from Snedden and Allen : School Reports and 
School Efficiency, pp. 118-127 

33 



m 



34 



School Statistics and Publicity 



Buildings ^| 

Is the number of sittings adequate? WM 

What is the value per child that can be comfortably seated? " 

How much has the city spent for buildings per $1000 of taxable 
property as compared with other cities for a period of years? 

How much waste space is there in the buildings? 

What is the proportionate cost of upkeep and repairs, per capita 
average daily attendance? 

What has been the cost of equipping special buildings or rooms, as 
laboratories, manual training, etc., per capita average daily at- 
tendance, as compared with other years and other systems? 
Receipts and expenditures 

What has been the per capita expenditure, expressed in terms of 
average daily attendance of the system as a whole for several years? 

What has been the per capita cost, per school, of such items as 
salaries, general administration, fuel, building and repairs, 
special classes, etc.? Also what have been the relative costs? 

What do the different classes of schools, — high, junior high, ele- 
mentary, special, — cost per capita? 

Are receipts keeping pace with the increase in number of children? 
Census 

What are the numbers of children for each year within the school 
age limits in the city ? 

How many of these are within the compulsory school age ? 
Attendance 

What is the number of children of compulsory school age attend- 
ing public schools? 

How many children of compulsory school age are attending private 
schools, special schools, or are otherwise satisfactorily accounted 
for? 

How many are in voluntary attendance, per school, and has the 
number increased during the past several years? 

What is the number of persistent attendances, i.e., children who 
attend 160 days out of a possible 200, etc.? 

What is the character of the absences? 
Elimination 

How many children drop out each year, by grades and schools? 

Why do these children drop out — transferred to other schools, 
non-promotion, irregular attendance, over-age, etc.? 

Is the percentage of elimination increasing? 



Collection of Data 35 

RetardoMon 

What are the percentages of promotion and non-promotion by- 
grades and schools? 

Would retardation be lessened by flexible promotion schemes? 
By promotion by subject in the upper grades? 

Are retardation and elimination increased or lessened in schools 
where industrial work is given? 

What is the relation between retardation and over-age? 
Special classes 

What is the per capita cost, average daily attendance, of special 
classes, evening schools, etc.? 

How does this cost compare with that in other cities? 
Medical inspection 

How many children have been treated ? How many defects have been 
remedied as a result of this treatment ? How many homes visited ? 

What is the per capita cost of this work ? 

Is it adequate? 
Truancy 

How many cases were reported? 

What disposition was made of these cases? 

What were the causes of these cases of truancy? 

What has been the per capita cost of the truancy department ? 

How does it compare with other cities in amount of work done and 
in expense? 

How m.uch time on the average elapses before a reported truant is 
returned to school ? 
Supervision 

Which of several methods of teaching a subject produces the best 
results with the given teachers and conditions of the system? 

Which teachers or schools are doing the best work, in what subjects, 
and just how much better? 

How many teachers can a supervisor most profitably work with in 
the given system? 

How successful is the scheme of the system for rating teachers ? 

How effective for producing better teaching are the methods em- 
ployed for improving teachers in service? 

Are the teachers overburdened with routine and clerical work? 

Are the classes too large for good work? 

Which parts of the curriculum ought to be eliminated or given re- 
duced time allotments? 



36 School Statistics and Publicity 

2. When Statistical Method Is Unprofitable 

Sometimes statistical method cannot be used, because 
it is impracticable to get suitable data. Lack of records, 
of suitable help, of money to pay for such assistance, or of 
time on the part of the superintendent, may bring this about. 

There are cases also in which the effect aimed at with 
a statistical presentation could be secured more quickly 
and easily with some argument from analogy, some emo- 
tional appeal, etc. For example, one of the Cleveland 
reports some years ago wished to emphasize the improve- 
ment in school children that could be produced by op- 
erations for adenoids and similar troubles. Instead of 
using statistics, it simply showed the picture of a boy 
afflicted with adenoids before the operation, and another 
picture of him some months afterward. The difference 
in expression in the two pictures, especially as regards 
intelligence, probably was more effective than any 
possible statistical presentation. 

Then there are cases of individuals which, as Professor 
King so aptly puts it, " statistics cannot and never will 
be able to take into account. When these are important, 
other means must be used for their study." ^ The second 
of these statements is really true only on the assumption 
that the superintendent has the right conclusion to present. 
Often this statistical treatment would show him that his 
presentation was untrue or at least needed material 
modification. 

Finally, there are some school problems that cannot be 
treated statistically with profit, for example : 

Cost of subjects sure to enroll few students compared with the cost 
of those sure to enroll many. 

Getting a city already spending money freely to do better. 
1 King, W. I. : Elements of Statistical Method, p. 35 



Collection of Data 37 

Selection of texts ; cheap books usually mean inferior quality. 

Percentages of elimination and retardation where it is impossible 
to estimate fairly accurately the number of children entering in a 
given year. 

Worth of teaching methods outside of those that may be objec- 
tively measured. Superintendent Maxwell in the article previously 
cited gives a few such cases, for example, the items of character- 
building, development of reasoning ability (not yet measured to any 
appreciable extent), the motive to good, hard work, and all problems 
involving tact.^ 

3. How to Decide Doubtful Cases 
How shall the superintendent decide quickly which 
problems shall be statistically treated and which not? 
Conditions vary often with each problem. But in 
general, if the big elements in a given problem involve 
numbers or can be expressed or measured in numbers, 
statistical treatment will be applicable. The cost of 
subjects sure to enroll few students cannot profitably be 
statistically compared with the cost of subjects sure 
to enroll many students, because the big factor in the 
situation cannot be stated in numbers. This factor is 
the relative value of the two subjects as parts of educa- 
tion. One may correspond to salt, of which all of us 
need a few grains each day, preferably some at each 
meal, and for which there is no substitute. English is a 
good example. The other may correspond to protein, 
which all of us also need, but which does not have to be 
taken at every meal, or even every day, and of which 
there are various forms. Science with its numerous 
alternative forms is a good example. Each subject is as 
important as the other for perfect educational health, 
but this relative importance can hardly be profitably 

1 Maxwell, W. H. : "How to Determine the Efficiency of a School 
System," American School Board Journal, May, 1915, p. 11 



38 School Statistics and Publicity 

treated or measured with numbers. Before rushing 
into any statistical treatment of a school problem, the 
superintendent, then, should first try to analyze out the 
big factors, considering whether the most important ones 
can be satisfactorily treated with statistical method. 

EXERCISES 

1. Which of the following problems, or which parts of them, may 
be profitably subjected to statistical treatment? Which may not? 
Give your reasons precisely for each one. 

(a) To what extent can science be profitably taught in the 
grades ? 

(6) What percentage of the funds available for library pur- 
poses in a high school should go to each department in 
a given year? 

(c) Should home teachers be paid markedly lower salaries than 

those from a distance? 

(d) Should the passing mark be 65 or 70 on a scale of 100? 

(e) Should a given child in a given grade be promoted? 
(/) Is a given teacher marking too hard or too laxly? 

(g) What salary as superintendent in a given state may a com- 
petent man reasonably look forward to? 

2. State in the form of definite questions at least three school 
problems in which you are interested that might profitably be subjected 
to statistical treatment. Give your specific reasons for so listing them. 

3. Do the same for at least three school problems in which you are 
interested that cannot profitably be subjected to statistical treatment. 

II. HOW TO PLAN STATISTICAL TREATMENT OF 

PROBLEMS 

Careful planning in statistical work is always a sine 
qua non for success. " Each hour spent in carefully 
arranging the work is likely to save a score of hours in 
trying to straighten out the confusion due to a hasty and 
ill-advised program." ^ It is equally true that " one 
^ King, W. I. : Elements of Statistics, p. 47 



I 



Collection of Data 39 

of the peculiarities of statistical work is that practically 
everything must be anticipated in advance, all possible 
sources of error detected and guarded against, and even 
the general results estimated." ^ The saving of time in 
statistical work becomes all the more necessary when we 
remember that the superintendent is at best a very much 
overworked man. Careful planning means time saved 
for the really big things in the statistical work he does, — 
the results and their meaning for his school work. He is 
also able to get a larger number of statistical processes 
done in a given time, which in turn means that more 
results and meanings will be available. In a word, 
careful planning of statistical work permits a larger use 
of statistical method by the superintendent in the time 
at his disposal. 

To save time, the main cautions to be kept in mind in 
planning statistical work will be given briefly and dogmat- 
ically with only necessary explanation. 

1. Decide Precisely What Is to Be Found Out or Proved 
in the Statistical Work 

Indefinite phrasing of the problem means indefinite 
thinking, with the inevitable wastes of time that accom- 
pany it. The best device the author has ever found for 
compelling one to make a sharp and clean-cut statement 
of the problem, is to state it in the form of a very definite 
question, the adjectives, adverbs, subordinate phrases, 
etc. of which indicate the sub-questions or minor problems. 
This device has been found to be very serviceable on many 
problems not involving statistical treatment. It is 
just as serviceable on those that do need statistical 

1 King, W. I. : Elements of Statistics, p. 47 



40 



School Statistics and Publicity 



method. The usual topical statement of the problem is 
one of the surest guarantees of loose thinking at the 
start. The interrogative form of statement accentuates 
the problem effect. A superintendent may simply state 
his problem as '' School Costs in Blankville." How much 
better it would be to state it thus : '' Are we paying all 
we can possibly afford for schools and are we getting our 
money's worth? " The same thing holds true for sub- 
ordinate problems. For example, compare the ordinary 
superintendent's statement of his problem with that of 
Superintendent Spaulding in the 1912 school report for 
Newton, Massachusetts. 



Usual Statement 


Superintendent Spaulding's 
Statement 


Report of Blankville Public 

Schools 
Statement of aims 


The Newton Schools : what are 
they trying to do? 


Attendance and progress of 
pupils 


Are they doing what they are 
trying to do ? 


(Taken for granted) 


Do you approve of their policy? 


Expenditures of the system for 
the current year 


Is their policy carried out eco- 
nomically? 


Administration 

Course of study 

Reports of various supervisors 


Is it administered efficiently? 


Recommendations for next 
year's work 


Can we afford to continue it? 
Can we afford not to continue 
it? 



Collection of Data 41 

2. Plan to Collect only Data for Which One Can Point 

Out in Advance Specific Ways in Which They 
Will Be of Value to Him 

This does not of course mean that one can know in 
advance all the ways in which the data will be of value 
to him. The collection of data that do not seemingly 
answer some of one's problems or promise to buttress 
certain of his arguments, is simply a gambling proposition. 
And the odds are ten to one that the data will never be 
of any material use to him. It is true that sometimes 
unexpected uses for data will appear after the work of 
collecting them has begun. For example, the writer and 
a graduate student collected considerable data on the 
cost of instruction in southern normal schools. But they 
had worked for some weeks before they discovered that 
the same data would give material for answering very 
important questions about size of classes. However, 
such a valuable by-product cannot be counted upon in 
all cases. 

3. Plan the Whole Procedure Through to the End, 
Trying It Out on Sample Data to Be Sure That the 

Units, Blanks, Processes, etc. Will Work 

Here is the place where one hour of good work will save 
twenty later on, as Professor King says. The units 
chosen should be carefully tested to see if they are prac- 
tical. The blanks should be drawn out in detail and the 
actual operations attempted with them. 

In particular, what is known as the question of " group- 
ing " must be decided. This means that if the data 
are to be considered in groups, the exact range of each 
group must be determined beforehand. For example, 



42 School Statistics and Publicity 

if one is studying days' attendance, are the children to be 
grouped as those attending 0-19, 20-39, 40-59, etc., or 
0-9, 10-19, 20-29, etc. ? This cannot be treated in detail 
here, but is discussed fully on pages 106 and 107. 

If the blanks are to be filled out by outside persons, 
some of the actual people, or preferably similar but less 
intelligent people, should be used to test out the blanks. 
The errors that these persons make should be noted and 
the blank revised accordingly. Thus, in sending out a 
blank for teachers to fill out, it is advisable to submit the 
rough draft to several average teachers, see how they can 
fill it out, and revise as necessary. 

A very helpful device at this point is to make a " brief " 
of just what is to be found and of the methods to be used. 
This can be elaborated from the material accumulated 
under the suggestions in i and 2. 

The actual processes for handling the data should be 
tested on the blanks themselves. Thus in making out 
a blank, if percentages are later to be calculated, the 
numbers from which they are to be calculated should come 
in adjacent columns if possible. Actually calculating 
such percentages on sample blanks will insure that an 
economy of this nature is cared for. Again, in making 
out blanks in series, the same fact should appear if possible 
in relatively the same column in different blanks. This 
will make all manipulation and calculation much easier. 
But such placing is almost sure to be overlooked unless 
the calculation with simple data method is carried through. 

In all probability the reader by this time is asking this 
question : " But how can one keep open-minded if so 
much planning is to be done ahead? " This is a natural 
question. So is the usual one as to whether such planning 
will not tend to the buttressing of preconceived opinions 



Collection of Data 43 

rather than to the discovery of anything new. In the 
attempt to find the truth about anything, the question 
method of outlining the plans is undoubtedly the surest 
mechanical device to aid in keeping one open-minded. 
Note that Superintendent Spaulding does not suggest 
the answer to any one of his questions. Beyond this, 
probably no device will be of much service to a person 
who is determined to prove a certain thing by statistics 
whether or no. 

Furthermore, much of the superintendent's statistical 
work is for the purpose of demonstrating or proving to 
others what the superintendent already knows to be 
true. Here the element of keeping open-minded does not 
enter, and planning ahead is unquestionably very helpful. 

EXERCISE 

Take any one problem from your list in Exercise 2, page 38. Out- 
line in question form precisely what you would wish to find or prove. 

III. HOW TO DETERMINE UNITS AND SCALES 

In planning statistical work, the need of units and 
scales early becomes apparent. ^ The superintendent is 
constantly called upon to pass judgment upon the worth 
of many school matters. He generally does this by 
merely placing the thing judged in its proper place in a 
gi'aduated scale of values of such things. For example, a 
superintendent passing judgment on the work of a teacher 
merely puts her in her proper place in the list of teachers 
in his school system, ranged from high to low, or in a list 
made up of all the teachers he has ever observed. This is 

1 See pp. 3-11 for examples of errors arising from a lack of proper 
units and scales. 



44 School Statistics and Publicity 

indicated in the very language he uses : '' Best I ever 
had/' *' worst I ever saw," " hopeless," '' practically 
perfect," etc. If he expresses his judgment in letters, 
as A, B, etc., or in figures as 85, 90, 100, etc., he is merely 
substituting symbols for such word estimates. 

1. Subjective and Objective Scales 

It is generally recognized that the superintendent 
will vary at different times in his judgments of the same 
teacher, engaged in the same kind of work. He may be 
suffering from a severe headache, or be perturbed over a 
recent business reversal. But his readings of a standard 
thermometer, when it was at the same temperature, 
would vary little from time to time. 

Let us now consider the reasons for this difference. 
The thermometer is graduated into constant definite units 
that measure the same amount of heat in the room always 
in the same way. Not so in the case of judging the 
teacher. The superintendent's scale of teaching ability 
has no definite units that always measure the same amount 
of teaching efficiency. 

These two scales represent the extremes of the kinds of 
scales that the superintendent must use. The units in 
the scale for judging teachers are in the superintendent's 
mind. Granting that he can transmit fairly clear ideas 
of his scale to others, there will be great disagreement 
among those using it. If they agree, they may easily 
be unaware of the fact, for the same descriptive words 
mean very different things to different persons. Since in 
a scale of this kind there are no units that can be made to 
mean exactly the same thing to different people, such a 
scale is said to be a '' subjective " one. In the case of 



Collection of Data 45 

the scale for measuring temperature, the units are not 
concealed in the mind of the person using the scale; 
they are external to every one who wishes to use it. For 
this reason, such a scale is called " objective." The 
chances for error or difference of opinion in reading 
the units of an objective scale are slight as com- 
pared with those arising from the use of a subjective 
one. 

Between these two tjrpes of scales lie others with vary- 
ing degrees of subjectivity and objectivity. For example, 
the Thorndike handwriting scale is made up of samples 
of handwriting rated from to 18, as determined by the 
combined judgment of a considerable number of com- 
petent judges. This scale is objective in that any one 
can see the sample of handwriting grades, say No. 8. 
But its use is subjective in that all people do not agree 
that No. 8 is worse than No. 9, nor would all judges rate 
any other sample of handwriting at the same place on 
the scale, say No. 12. If the number of judges were 
large enough, the variation in placing such a sample 
might range from 10 to 17. 

It is highly desirable in planning any statistical work 
to try to secure units and scales that shall be as objective 
as possible and that shall have a minimum of harmful 
subjective elements. 

Thus, the step between 94 and 95 in the marking 
system of two teachers does not mean at all the same 
thing. Again, merely because several people think two 
words are equal in difficulty, it does not make them so. 
Professor Thorndike, as noted on page 9, quotes Rice as 
counting that disappoint is equal to feather in difficulty 
in spelling, or as proceeding as though it were. But by 
actual experimentation in a 5A grade, twenty-four times 



46 School Statistics and Publicity 

as many girls and thirteen times as many boys missed 
disappoint as missed feather. In measuring arithmetic 
work, it is much better to take examples from the Courtis 
or Stone tests, because the practical worth of these has been 
demonstrated by the actual achievements of thousands 
of pupils. 

One reason why subjective scales are so often un- 
desirable is that the zero points on them are unknown. 
On an objective scale, such as length, 90 inches is just 
three times as long as 30 inches, or it is just three times 
as far from zero length. But in the grading or giving of 
marks by two teachers, there is no assurance that each 
regards 90 as just three times as good as 30 or just three 
times as far removed from utter failure. One teacher 
on a test may grade the worst student in the class at 30 
and the best at 90. Another teacher might grade these 
same students on the same test at 60 and 90. The teachers 
would obviously be grading from different zero points. 
In the standard scales for grading composition, hand- 
writing, and so on, the zero points have been determined 
by a procedure too complicated to be given here.^ This 
is why such scales, even when they involve many sub- 
jective elements, are so superior to the attempt of a 
novice at making his own scale. 

If we must take subjective estimates as units and make 
our own scale, it is better to pursue the following treat- 
ment : 

1. Avoid choosing estimators with known or probably marked 
prejudices. 

2. Have all these persons estimate the worth of the problems in 
terms of a separate problem, which for convenience is to be consid- 

1 See Thorndike : Mental and Social Measurements, Revised Edi- 
tion, p. 16^. 



Collection of Data 47 

ered worth so much, say 10. (That is, all call this problem the value 
of 10.) 

3. For the value of any other problem, take the average of the 
estimates given by the different persons.^ 

2. The Jingle Fallacy 

The superintendent must beware especially of con- 
sidering things equal because they are called by the same 
words. This is known as the " jingle " fallacy.^ Thus, 
one child does not equal another child as a matter of 
school expenditure, if the first child is in the primary 
grades and the second child is in the last year of the high 
school. The cost of educating the latter for one year is 
much more than in the case of the former. The differ- 
ence between the ability to do one problem and the 
ability to do two problems in the Courtis tests is not the 
same as the difference in ability to do fifteen problems 
and the ability to do sixteen problems. Any one who 
can do fifteen can fairly easily work up to sixteen. But 
if a child can barely do one, it is a tremendous task to 
work up to doing two. The '' jingle " fallacy usually 
results from neglecting to define units or to consider the 
zero points. 

3. Essentials of a Valid Scale 

The construction of a good scale for many lines of 
school work demands considerable technical knowledge 
and experience. The superintendent in general had 

1 Professor Thorndike on pages 9 and 10 of Mental and Social 
Measurements has a much more complicated method for utilizing sub- 
jective estimates. 

2 Professor Thorndike borrows this term from Professor Aikins. 
See Mental and Social Measurements, p. 10. 



48 School Statistics and Publicity 

better plan to use scales already worked out.^ Beyond 
this we may for our purposes summarize the essentials 
of a valid scale from Thorndike : ^ 

1. The scale must be as objective as possible. 

Its meaning must be such that all competent judges will agree 
as to what it is. 

2. The series of facts used in making up the scale must be of the 

same sort of thing or quality. 

3. The steps in the scale should be clearly defined. 

It is better if they are equal ; if unequal, let the steps be de- 
fined as definitely as possible. However, a scale in which 
only the order or rank of the various facts making it up is 
known, is often very useful. 

4. The zero point must be defined if possible. ^ 

4. Discrete and Continuous Series 

It is impossible to use a scale properly unless one knows 
whether the facts it is to measure are in a discrete or a 
continuous series. A series is said to be discrete if it is 
regarded as broken up, i.e., the different items are separate 
or there are gaps between them. On the other hand, if 
the series is capable of any degree of subdivision, that is, 
if the items are regarded as strung out along the scale, 
and running into each other, the scale is said to be con- 
tinuous. The table of the costs of instruction in mathe- 
matics, page 18, is an example of a discrete scale. In this 
table every item is regarded as an integer and there are 
gaps between the items. 

A good example of a continuous series is Table 2, made 
up from data worked out by the writer. 

iSee "Descriptive List of Standard Tests," by W. S. Gray, Ele- 
mentary School Journal, 18: 56. (Sept., 1917.) 

2 Thorndike, E. L. : Mental and Social Measurements, pp. 11-18. 



Collection of Data 49 

Table 2. Continuous Series Showing Fifth Grade Achieve- 
ments WITH Courtis Tests in Addition, in a Western City 



Number 


of 


Number of children 


>lems attempted 


making each score 


1 







2 




2 


3 




10 


4 




24 


5 




32 


6 




35 


7 




34 


8 




54 


9 




25 


10 




27 


11 




24 


12 




4 


13 




4 


14 




3 


15 




6 


16 




9 


17 




2 


18 




2 



In such a series as this one, the 32 children who at- 
tempted five problems are not all regarded as being pre- 
cisely at the point " five problems attempted " on the 
scale, but as distributed from " five problems attempted " 
to " six problems attempted/^ For this reason it is 
very important to know what a given number means 
on a scale. That is, does 6 mean from 5.5 to 6.5, or from 
6 to 6.99, or nearer 6 than either 5 or 7? The second of 
these methods, that of measuring in terms of the point 
last passed, is often the natural way and saves labor in 
all sorts of measurements. ^ This method is the one to use 
where it is possible to say authoritatively that a given case 

1 See Thorndike : Mental and Social Measurements, p. 22. 



50 School Statistics and Publicity 

is beyond a certain point on the scale, but " the how much 
beyond " cannot be easily determined. Obviously it is a 
good method for Table 2, since a given case has attempted 
say five problems, but one cannot easily tell whether it has 
just started on five, is half way through the fifth, or is 
practically ready to start on the sixth. The other method 
can be used with a scale like the hand- writing scales, where 
a given case is said to be nearer a given sample on the 
scale than anything else. A case would be called "9,'^ 
for example, without being definitely located as either 
below or above that point. Here '' 9 " would mean from 
8.5 to 9.5. 



5. How to Use Scales 

In actual practice the superintendent can measure the 
worth of his work in whole or in part, on one of three 
kinds of scales : 

a. He can 'place the thing measured in its relative position 
in a scale of items (school systems, rooms, classes, etc.), 
all considered from the same viewpoint and without the 
use of units. 

This is the method used by the superintendent or school board of 
Blanktown, when he announces that his town has the best schools 
in the state, or that So-and-So makes this statement. It of course 
carries no weight whatever unless we know that the judgment of the 
one making the statement is sound. If one could read all the small 
town papers of any given state for one year, he would probably find 
three-fourths of them claiming that their home town had the best 
schools in the state. The same method with all its weaknesses is 
often used by a teacher in regard to the value of his pet method 
of teaching, his favorite mode of discipline, or the particular class 
he happens to be teaching at this time. 



Collection of Data 



51 



Table 3. Ranks of Certain Cities on Real Wealth and 

Assessed Wealth behind Each $1 Spent on Schools 

(Adapted from* Portland Survey, pp. 80, 304) 





Real Wealth 




Assessed Wealth 




City 


behind Each 
$1 for Schools 


Rank 


behind Each 
$1 for Schools 


Rank 


Newark, N. J. ... 


$165 


1 


$165 


11 


Worcester, Mass. 




180 


2 


180 


15 


Toledo, 




184 
185 


3 

4 


110 
185 


2 


New Haven, Conn. . 




18 


Paterson, N. J. . . 




185 


5 


185 


19 


Lowell, Mass. . . . 




194 


6 


194 


20 


Fall River, Mass. 




196 


7 


196 


21 


Syracuse, N. Y. . . 




202 


8 


180 


17 


Cambridge, Mass. 




204 


9 


204 


22 


Grand Rapids, Mich. 




207 


10 


166 


12 


Dayton, 0. ... 




208 


11 


125 


4 


Washington, D. C. 




212 


12- 


148 


8 


Scranton, Pa. . . 




216 


13 


173 


18 


Jersey City, N. J. 




218 


14 


218 


24 


Columbus, 0. . . 




221 


15 


133 


6 


Rochester, N. Y. . 




225 


16 


180 


16 


Denver, Colo. . . 




231 


17 


116 


3 


Albany, N. Y. . 




234 


18 


234 


26 


Providence, R. I. 




256 


19 


256 


31 


Bridgeport, Conn. 




276 


20 


276 


34 


Kansas City, Mo. 




280 


21 


140 


7 


Minneapolis, Minn. 




294 


22 


132 


5 


New Orleans, La. 




314 


23 


236 


27 


Louisville, Ky. 




326 


24 


228 


25 


Nashville, Tenn. . 




350 


25 


263 


33 


Omaha, Neb. 




352 
354 


26 

27 


53 
177 


1 


Oakland, Cal. . . 




14 


Seattle, Wash. . . 




364 


28 


164 


10 


Spokane, Wash. . 




370 


29 


152 


9 


St. Paul, Minn. . 




407 


30 


204 


23 


Indianapolis, Ind, 




408 


31 


245 


29 


Memphis, Tenn. . 




449 


32 


247 


30 


PORTLAND, ORE. 




456 


33 


260 


32 


Birmingham, Ala. 




479 


34 


240 


28 


Richmond, Va. 




536 


35 


402 


35 



52 School Stat'* xs and Publicity 

A refinement of the same method is used by a judge in a contest, 
when he ranks the contestants in order of merit only. He then 
gives the best contestant the rank of l,.the next best the rank of 2, 
and so on. But he wisely refrains from attempting to say how much 
better is the first.^ 

h. He can compare his own school with other schools on 
a scale of his own making, all schools being measured with 
definite units. 

Thus, he may wish to compare his community with others on the 
basis of the amount of money it really pays, wealth considered, for 
schools. The Portland Survey Commission found in comparing the 
wealth and school expenditures of that city with thirty-six others 
nearest it in size, that there was a vast difference between the 
assessed and the real wealth in many cities. To show this point 
more clearly the table on p. 51 has been adapted from the Portland 
Survey. 

It may be seen from this table that it is very important to have as 
the unit, the number of dollars of real wealth in the city, not the 
number of dollars of assessed wealth. If assessed valuation were 
taken as the unit, Omaha would be 1 instead of 26 ; Denver 3 instead 
of 17 ; Lowell 20 instead of 6, etc. 

It is evident that any one seeing this or a similar scale must agree 
to the ranking of each city or item as shown in the first column, 
provided the original data for the calculation are correct and the unit 
is a reasonable one. Then there remains only the question as to 
whether the cities selected for the scale are representative ones for 
fair comparison in the matter under discussion. 

c. He may compare his own school with other schools hy 
means of a standard test, and then place his school on a 
scale of cities made up as in h, or merely compare it with 
the standards of the makers of the test. {This amounts to a 
scale.) 

The advantage of a scale of this kind is that the units have been 
proved equal or approximately equal, and there can be no question 
of the relative positions of samples in a given scale. And as time 

1 See pp. 11, 192-198 



I 



Collection « ' Data 53 

goes on, very authoritative scales will appear. Thus, we now have 
scales of this sort in the Thorndike or the Ayres handwriting scales, 
the Hillegas or the Harvard-Newton composition scales, the Ayres 
spelling scale, etc. Furthermore, by taking the achievements of 
school systems as measured on these scales or with standard units, a 
superintendent can easily make a scale of such achievements and 
see where his school system comes on it. 

6. Practical Examples of Units and Scales for 
Superintendents 

The best practical examples of units and scales for a 
superintendent, of course, appear in the recent school 
surveys. A superintendent wishing to get up a scale 
or find units on a given problem can get them very quickly 
by utilizing the following table. The particular survey 
is denoted by the name of yie city, and the numbers 
refer to pages. 

Units and Scales that a Superintendent may Profitably Use 

Description Where found (Surveys) 

Playgrounds 

r Salt Lake City, 222 
Square feet per child < Rockford, 7 ; Ashland, 11 

I Denver, 11, 122 
Buildings 

Square feet per child ( f " ^'^,?"^' ''' 

t Ashland, 11 

Cubic feet per child Ashland, 11 

Space per teacher and child Leavenworth, 48 

Number of sittings per room Snedden and Allen, 29 

Total seating capacity by 

buildings Snedden and Allen, 29 

. . I,- r ^ f San Antonio, 315 

Average cost per cubic foot l ^ . ^ , , Ar. no 

\ Sprmgiield, 22, 23 

Average cost per pupil San Antonio, 315 

Average cost per classroom San Antonio, 315 

Same for fuel Denver, 1, 55 



54 



School Statistics and Publicity 



Same for repairs 
Lighting by candle power 

Janitor's salary per room 

Same per hour 

Same per 1000 cubic feet 

Valuation per room 

Teaching staff 

Number of children per super- 
visory officer 

Same on average daily at- 
tendance 

Number of children, average 
daily attendance per teacher 

Training of teachers 
Years of experience 
Years of training 

Teachers' salaries 



On yearly basis 



On monthly basis 

On weekly basis 

Based on enrollment 

Based on years taught 

Maximum and minimum sal- 
aries 

Principal's salary based on 
number of rooms in building 
Proportionate expenditures 

Percentages of school expendi- 
tures for different purposes 



Denver, 1, 62 

Salt Lake City, 235 
/ San Antonio, 249 
\St. Louis, IV, 117-120 

Ashland, 11 

Ashland, 11 

San Antonio, 250 



Salt Lake City, 39 

Oakland, 26 
f Salt Lake City, 53 

Louisville, 33 

Newton, 1913, Table IX 
. Oakland, 24 

South Bend, 198 
South Bend, 200 

f South Bend, 101 

Leavenworth, 50 

Butte, 120 

Newton, 1913, V 

Janesville, 43, 44 

Bridgeport, 17 

Vermont, 225 
. Springfield, 61 

Ashland, 14 

Vermont, 225 

Baltimore, 74 

Salt Lake City, 55, 56 



Portland, 75 
f Cleveland, 97 
\ Springfield, 61 

I Janesville, 74 

\ St. Louis, IV, 55, 56 



Collection of Data 



55 



Percentage of salaries for high 

schools 
Percentage of city expenditures 

for schools 

Per capita costs 
Total population 



For each of population over 15 

For each adult male 

High school costs per person 

in population 
Average daily attendance 

Same for fuel 

Enrollment 

Average number belonging 

Cost of instruction 
Student hour 



Per pupil 

Per pupil enrolled 

Per 1000 student hours 

Miscellaneous costs 

Expenditures for whole cities 

on medical inspection 
Evening schools per session 
Per wagon, rural consolidated 

schools 
Part of each $1000 spent on 

instruction in each subject 



Janesville, 82 

J Janesville, 68 
\ St. Louis, IV, 32 

- Butte, 143-4 

Bridgeport, 21 
" Janesville, 68 

Oakland, 44 
I Baltimore, 34 

Portland, 407 

Portland, 407 

Springfield, 95, 96 

In most surveys 
' Butte, 82 

Oakland, 44 
I Kansas City, 82 
[ Birmingham, 36 

Houston, 83 

Sneddon and Allen, 17 

' South Bend, 204 
Vermont, 227 

I Leavenworth, 51 

. Springfield, 114, 97 
/ Rockford, 111 
\ Janesville, 75 
Springfield, 114, 97 

r San Antonio, 215 
\ Denver, 1, 60 

I Janesville, 90, 100 



South Bend, 177 
Newton (1913), 41 

Texas, 33 

San Antonio, 213 



56 



School Statistics and Publicity 



Expense for attendance officer 
per 1000 pupils enrolled 
Time spent on each subject 

Minutes per week 

Part of each 1000 hours spent 
on each subject 

Hours of recitation and di- 
rected study in reading and 
history 
Population 

Number per 1000 in certain 
occupations 

Same by lOO's 

Races by lOO's 

Family for nearly everything 

Wealth 



Portland, 390 

f Salt Lake City, 76 
I Leavenworth, 54 
[ Houston, 83 

San Antonio, 214 
Cleveland, 121, 125 



South Bend, 141 
- r Salt Lake City, 17 ' 
. \ Cleveland, 21 

Cleveland, 21 

Red River, 42, 48, 82 



Per capita population 
Same for real wealth 



r South Bend, 211 
< Janesville, 68 
[ St. Louis, IV, 18 
r Salt Lake City, 19 
1 Cleveland, 25 
Taxable wealth behind each / Oakland, 43 

child in school 
Same for child in average daily 

attendance 
Real wealth behind each $1 

spent on schools 
Possible revenue per child en- 
rolled 
Tax rate 

Mills on assessed valuation 



1 Maryland, 128 
Janesville, 52 
Portland, 414 
Salt Lake City, 48 



Same on real valuation 



Same per $100 real valuation < 



Illinois, 262 
r Portland, 108 
\ Leavenworth, 17 
' South Bend, 215 

Janesville, 60 

Salt Lake City, 313 

Rockford, 119, 49 



Collection of Data 57 

Per $100,000 real wealth Cleveland, 25 

Rate necessary to produce esti- 
mated per capita support for 

schools on actual wealth Salt Lake City, 311 

Enrollment 

Increase in children per week 

(5 years) Salt Lake City, 36 

On basis of 1000 children in 
kindergarten Ogden, 9 

Parts of 100 pupils in public, 
private, and parochial 
schools Cleveland, 28 

Same, failures, and promotions Denver, 1, 70, 2, 77 

EXERCISES 

1. Discuss the value for the superintendent of the units used in 
each of the examples given on pages 53-57, and of the scales that 
could be made up from such units. Just how would you make up 
these scales? 

2. What is the value of the question for measuring the efficiency of 
teachers, counting the number of questions they ask in a given time ? 
Why? 

3. Which is better for measuring the preparation of high school 
teachers, and why? 

(a) The number of years they were in college. 
(h) The number of years beyond the elementary school they 
spent in study. 

4. Precisely what is the value of each of the following methods of 
instructing judges in a contest, and why? 

(a) Mark on a scale of 100. 
(6) Mark on a scale of 30. 

(c) Mark the best 1, the next best 2, etc. 

(d) Mark the best 100, the worst 70, and the others where 

they should come in between. 

(e) Mark on a scale of 100, allowing 50 for content and organ- 

ization, 30 for English, and 20 for delivery. 

5. What units and scales would you plan to use in studying the 
statistical problem selected in the exercise on page 43, and just why? 
Precisely how would you plan to use the scale or scales chosen ? 



k 



58 School Statistics and Publicity 

IV. HOW TO DO THE ACTUAL COLLECTING 
1. Records in One's Own School 

Most of the superintendent's data must come from his 
own school system. But on many problems the matter 
of giving advice about the collecting may be like that of 
Holmes, when he said that one should always exercise 
great care in the selection of one's grandfather. The 
superintendent can at any rate collect far more data by 
seeing that his records are so kept as to show the desired 
facts later, than he can ever suddenly exact from teachers 
who have never thought of keeping or giving out infor- 
mation on this point. For example, the disputes and 
troubles in the studies of retardation and elimination a 
few years ago arose mainly because of the way school 
records had been kept. It was impossible to tell from 
existing records how many of the given pupils had entered 
the first grade at any given time years before, or to find 
all the significant facts about a pupil brought together 
in one place. 

Inasmuch as the facts in the superintendent's own school 
are often meaningless unless they can be compared with 
similar facts in other school systems, he must as far as 
possible use records similar to those of his fellow school 
men. Therefore, it is advisable that he use the records 
and reports recommended by the Committee of the 
National Education Association on Records and Reports, 
and by the United States Bureau of Education. He 
should also try to get the State Department of Education 
in his state to use blanks that will fit in with this system. 
If a decalogue for superintendents should be written, 
one of the first commandments undoubtedly should be: 
" Thou shalt keep thy records as nearly as possible by 



Collection of Data 59 

the uniform system of the National Education Associa- 
tion." 1 

2. Other Sources of Data 

The data from other school systems are obtained 
usually by the use of questionnaires, from printed reports, 
school surveys, magazine articles, etc. Aside from the 
question of selection, the questionnaire method is often 
practically worthless for collecting statistical data. 
School men are too busy to answer large numbers of 
questions, to work out the object of the questionnaire 
when this is not clearly stated, to hunt up much infor- 
mation on former conditions, to puzzle out involved or 
ambiguous questions, and sometimes too careless to give 
information definite enough to be of any service. As a 
result, few replies will come in on a given problem, 
and not all of these will be complete. This means few 
opportunities for comparison. 

It is always better to make use of printed statistics 
where possible, taking care to be sure of the units and 
processes used in compiling the statistics taken. The 
mere fact that they are part of a printed report or formal 
presentation, often required by law, practically insures 
much more accurate figures than can be secured by the 
questionnaire method. A second advantage is that the 
superintendent can see that all his figures are on the same 
basis, a thing impossible with a questionnaire because of 
the inability of many people to understand or follow 
directions on an inquiry which they have never seen 
before and never expect to see again. 

1 See Bulletin U. S. Bureau Education, 1912, No. 3, or get particu- 
lars from such companies as the Library Bureau or the Shaw-Walker 
Co. 



60 School Statistics and Publicity 



^ 



For busy superintendents the following suggestions as 
to where material for statistical comparisons may be 
found, are appended : 

1. Report of the United States Commissioner of Education, Volume 
II, each year contains numerous tables on city school systems. It 
includes such material as enrollment, number of teachers, aggregate 
days' attendance, average daily attendance for both elementary and 
secondary schools, length of session, number of children of census age 
in private schools, number of buildings, number of sittings, itemized 
receipts and expenditures of all school systems in cities of 5000 or more. 

2. The bulletins of the United States Bureau of Education often 
contain much valuable material on special topics. Especially good 
are such ones as the following : 

A comparative study of the salaries of teachers and officers. 
1915, No. 31. 

Ayres, L. P. : Provision for exceptional children in the pub- 
lic schools. 1911, No. 14. 

Boykin, Jas. C. and King, Roberta: The tangible rewards 
of teaching. 1914, No. 16. 

Deffenbaugh, W. S. : School administration in the smaller 
cities. 1915, No. 44. 

Frost, Norman : A statistical study of the public school sys- 
tems of the southern Appalachian Mountains. 1915, No. 11. 

Monahan, A. C. and Dye, C. H. : A comparison of the salaries 
of rural and urban superintendents of schools. 1917, No. 33. 

Morse, H. N. : Educational survey of Montgomery County, Md. 
1913, No. 32. 

Public, society, and school libraries. 1915, No. 25. 

Statistics of certain manual training, agricultural, and industrial 
schools. 1915, No. 19. 

Strayer, G. D. : Age and grade census of schools and colleges. 

1911, No. 5. 

Thorndike, E. L. : The elimination of pupils from school. 
1907, No. 4. 

Thorndike, E. L. : The teaching staff of secondary schools in 
the U. S. . 1909, No. 4. 

Updegraff, Harlan : A study of expenses of city school systems. 

1912, No. 5. 



1 



Collection of Data 61 

Updegraff, Harlan : Public and private high schools. 1912, 
No. 22. 

Updegrafif, Harlan and Hood, Wm. : Urban and rural school 
statistics. 1912, No. 21. 

3. The reports of the State Department of Education within the 
state where the superintendent lives. 

4. Strayer, G. D. and Thorndike, E. L. Studies in Educational 
Administration. (Contains the cream of many dissertations pub- 
lished at Teachers College prior to 1912.) 

5. Publications of the United States Census Bureau, especially 
special reports on cities and abstracts of each census. 

6. School surveys of all classes. 

7. Bulletins issued at various times such as those started by Super- 
intendent Spaulding at Minneapolis in September, 1916. 

8. Dissertations from leading schools of education in universities. 
The publications of the superintendent's own state university in this 
line will be easily available. In addition, the dissertations and theses 
from Teachers College, Columbia University, and the School of Edu- 
cation at the University of Chicago, are very valuable. 

9. Reports of investigations appearing in standard educational 
magazines. The best for this purpose are : 

American School Board Journal, Milwaukee 
Educational Administration and Supervision, Baltimore 
Elementary School .Journal, University of Chicago Press 
Journal of Educational Psychology, Baltimore 
School Review, University of Chicago Press 
School and Society, New York 

10. After the foregoing was written. Professor H. 0. Rugg's book 
on Statistical Methods Applied to Education appeared. It has much 
more extended references, especially on pages 28-39, 361-375. The 
material is admirably classified for ready reference. 

EXERCISE 

For the special problem you have selected, jot down as carefully 
as you can at this time : 

(a) The places in your own school system (or one in which you 
are interested) from which you might secure statistical 
data on it. 
(6) Other likely sources of statistical data on it. 



62 School Statistics and Publicity 

3. Sampling ^ 

As soon as the sources of data are determined, the 
question arises as to what can be done in the way of 
" sampling " with a view to cutting down the inevitably 
large amount of work. " Sampling/' of course, means 
working from selected or tjrpical specimens rather than 
with the whole mass of data. 

The superintendent usually considers sampling be- 
cause he must answer one of three questions : 

1. How many measures are needed of an item to be sure that the 
item is fairly well represented? 

2. How many cases and which ones need to be treated in a large 
mass of data to be sure that the results will be approximately true of 
the whole ? 

3. In case the superintendent at best can get only a small list of 
items for comparative purposes (say only a dozen towns that are 
really comparable to his town on school expenditures), how is he to 
choose these items? 

Let us now take these up in order. 

Number of Measures of One Item Needed. No 
arbitrary rule can be laid down for the number of measures 
needed on any one item. But it is safe to say that often 
there should be more than one in order to insure a reliable 
average measure for the item. For example, in comparing 
cities for school expenditures, it is often very unfair to 
get the expenditures for one year only. There may be 
very unusual conditions for that- year in several cities, 
say fires, cold spells that sent up fuel bills, epidemics that 
necessitated much medical expense, etc. In such studies 
it is customary to take the average of two years for each 
city, as Strayer does in his City School Expenditures, 

1 See also pp. 22-23 



Collection of Data 63 

Superintendent Spaulding, in a recent monograph on 
expenditures in Minneapolis, takes the average for five 
years for all cities studied. Another good example is 
seen in comparing teachers on their ways of marking 
students. It is very unfair to judge a teacher by the 
marks she gives at any one term examination, or in any 
one class. Professor Max Meyer at the University of 
Missouri, for instance, never attempted to pass judgment 
on the marking of a member of the faculty there until he 
could get at least five hundred marks given by that teacher 
in all his classes. A teacher may rightly object to a 
rating given her by the superintendent on one short 
visitation in one subject only. In practice, of course, she 
will not object to such rating if it is highly favorable 
to her, but it is probably about as far from the exact truth 
as an unfavorable rating would be if made on the same 
visitation. The writer some years ago, in inspecting high 
schools for the University of Missouri, found that the 
inspection was hardest in this one particular. It was im- 
possible to visit a school oftener than once a year for a 
day, and this made it extremely difficult to pass judgment 
on four or five teachers in the course of the six hours or 
less of teaching, judgment to which the teachers or their 
superintendents would really subscribe. 

The more measures we take of an item, provided they 
are not all chosen with the same bias or cause for error, 
the more reliable will be the final average measure taken 
for that item. But time is too valuable to permit going 
on indefinitely getting measures of one item. The safe 
procedure is to take as few measures of it as may reason- 
ably be expected to represent it fairly. 

Selection of Samples Ordinarily. The amount of 
work in any statistical treatment is so great that the 



64 School Statistics and Publicity 

question of cutting down the number of items by sampling 
is very important. But it is equally important that the 
samples be so selected as really to represent the whole. 
Neglect of this is responsible for the worthlessness of 
many laborious pieces of school statistics. 

The selection of samples should be absolutely at random, 
and if there are groups of data, the same percentage of 
samples should be taken from each group. Thus, a su- 
perintendent who was making a study of outside reading 
done by students in his town, could not get very trust- 
worthy results by asking the pupils in five wealthy families, 
in five families in comfortable circumstances, and in five 
poor families. There would be so many more in the 
second group than in the first, and so very many more in 
the third than in either of the other two groups that his 
results would be untrustworthy. To get at anything like 
the truth, he would have to take, say, twenty from the 
second group and possibly a hundred from the third. It is 
equally erroneous to attempt to obtain results on the effect 
of negro population on schools in Texas by ranging the 
counties in order of percentage of negro population and 
then taking three of the counties most free from negroes, 
three from the middle group, and three of those having 
the largest numbers of negroes. The results are wholly 
unreliable until we know how many counties are in the 
lowest natural group (that is, one without gaps in the 
percentages), how many in the middle group, and how 
many in the highest group, and that we have taken the 
same proportion of samples from each. Again, a teacher 
in giving a grade to a pupil ought not to take the work of 
the pupil during the last few days of the term. It is not 
a fair sampling of the student's work. Nor should 
she take the sample grades for her private book at any 



Collection of Data 65 

stated time known to students. If they know she always 
grades them on Friday, they will do well on that day and 
slow down on other days. 

Unfair selection explains why questionnaire methods 
of getting results are often so unsatisfactory and mis- 
leading. The people who answer questionnaires are 
often very selected and biased, and do not represent the 
whole group at all fairly. For example, a superintendent 
may wish to know what parents think of his school and 
may send out a questionnaire for this purpose. It will 
be sure to be answered mainly by those who are favorable 
or who wish to make him think they are favorable, and 
possibly by a very few opponents who are stirred up 
enough to come out frankly, but who are probably se- 
riously or unfairly prejudiced against him. If he sends 
out a questionnaire to other superintendents, as a general 
rule only the superintendents who think they can make 
a better showing than he can, will send in results. So 
very serious is this defect of the questionnaire method 
that the phrase of President Kirk of the Kirksville, 
Missouri, Normal School, " a questionable question- 
naire/' is often justified. The most casual examination 
of the statistical outpourings of the questionnaire type 
in the last few years will show that very often only a small 
percentage of the persons receiving questionnaires ever 
answer them. This small percentage is almost certain to 
be selected on some peculiar bias and is getting smaller 
because so many foolish and needless questionnaires have 
been sent out to superintendents in the last few years. 
Many school men confess to throwing nearly all question- 
naires into the waste basket or turning them over to 
clerks or pupils to answer. 

A good method of sampling is the familiar one of taking 



66 School Statistics and Publicity 

every fourth or fifth ease of the items arranged alphabeti- 
cally, or in order of magnitude, so that there can be no 
prejudice in the matter. This, of course, insures getting 
the same percentage in each of any possible groups. Thus, 
if it is a case of consulting citizens, every fifth name in 
the telephone book would give the superintendent a good 
sampling of men able to have telephones. But it would 
not represent all citizens. To get all represented, he 
had better take every fifth name in a directory of the 
city, or in a list of the registered voters. If it is a case of 
children, arranging them alphabetically by grades, per- 
haps boys and girls separately, and then taking every 
fifth name would give him a good set of samples for boys 
and another good set for girls. If a teacher has kept daily 
grades of pupils, he can get a good sample set of grades 
by taking, say, every seventh grade for each pupil, or 
something of this sort, just so long as the grade taken 
does not fall on the same day of the week every time. 

Sampling is often resorted to in giving standard tests 
to large numbers of children where the labor of grading 
the papers from all would be very great. One of the best 
sampling schemes coming to the writer's notice is the one 
employed in the San Francisco survey in 1916. Four 
tests — arithmetic, spelling, penmanship, and reading — 
were given. Two classrooms in different grades in the 
eighty-one elementary schools in the city were chosen at 
random for a test in some one of the four subjects. No 
teacher or principal knew in advance what rooms had been 
chosen or what subject would be given in the room selected. 

If a questionnaire has to be used, it is generally advisable 
for the superintendent to pick out a reasonably small list 
by one of the preceding methods, and then to devote his 
energies to seeing that approximately every name on the 



Collection of Data 67 

list sends in a fairly accurate answer. The " personal 
questionnaire " filled out in person by an '' interviewer " 
is sometimes an excellent device. It relieves the person 
interviewed of much drudgery and insures correct inter- 
pretation of the questions. But it tends to embarrass 
the person giving the information, often to the point of 
stopping easy conversation. To avoid this, the inter- 
viewer is apt to encourage or accept rough estimates in 
place of accurate data. In any case, the best way for the 
superintendent to get answers is first to be sure that his 
investigation will be of value to some one beside himself, 
and especially to those he requests to fill out his question- 
naire. Then he should promise to give all those answering 
it a copy of the results. If he ever intends to do any more 
investigating, he must faithfully keep this promise, 
aside from his moral obligation to keep it.^ 

Selection of Very Few Samples. Often the superintend- 
ent knows that he cannot by any possibility get more than 
a small number of items. His problem then is how to 
select these items so that he may be sure that his com- 
parisons will not be absurd. For example, if he wishes 
to know whether his community is spending enough 
money on its schools, is he to take all cities of that size 
in the United States or in his state ? Neither procedure 
will do because the same conditions manifestly do not 
obtain in all of these cities. The same would be true if 
the cities were selected on the basis of the number of 
children of school age, the number of children in school, 
etc. The towns certainly ought not to be compared 

1 For those who, in spite of the preceding, find it necessary to use 
a questionnaire, pp. 40-56 of Rugg's Statistical Methods Applied to 
Education will be very valuable. This contains the most practical 
treatment of the questionnaire that the author has yet seen. 



6S 



School Statistics and Publicity 



mercilessly unless their wealth per capita is something 
like the same. They cannot be compared on current 
school expenditures unless we know that their school 
debts are something like the same relatively, and even 
this may not be enough We may have to know their 
present taxes for other needs, the city's indebtedness for 
other purposes, etc. 

Mr. F. 0. Seymour in writing his master's thesis at 
Peabody in 1916 had the same problems in making a 
study of school costs for Amarillo, Texas. Table 4 is 
adapted from his work. 

Table 4. Showing School Statistics for Certain Cities of 
ABOUT the Same Population and General Situation 





. 


Per 


Per 








Per cent 


Town 


Popu- 
lation 


cent of 
pupils 
to total 
popu- 
lation 


cent of 
pupils 

in 
school 


Wealth 

per 
capita 


City 

debt 

per 

capita 


City 

tax 

per 

capita 


of city 

revenue 

going to 

schools 


Coffeyville, Kan. 


13,687 


24.0 


86 


$635 


$62 


$30 


48.1 


Dennison, Tex. 


12,632 


19.4 


69 


501 


22 


14 


41.9 


Pearsons, Kan. 


12,363 


18.3 


71 


815 


31 


15 


52.9 


Sherman, Tex. 


12,412 


20.9 


71 


608 


20 


15 


39.1 


Guthrie, Okla. 


11,654 


20.1 


72 


474 


68 


14 


34.2 


Marshall, Tex. 


11,452 


19.4 


61 


409 


51 


13 


33.4 


Paris, Tex. 


11,269 


27.2 


85 


815 


51 


19 


36.5 


Palestine, Tex. 


10,480 


18.1 


59 


508 


19 


10 


40.3 


Cleburne, Tex. 


10,464 


24.3 


78 


508 


31 


14 


40.8 


San Angelo, Tex. 


10,321 


15.2 


49 


566 


20 


10 


44.8 


Amarillo, Tex. 


9,957 


16.8 


61 


549 


26 


9 


45.7 



It may be noted concerning these cities which Mr. Sey- 
mour has selected for purposes of comparison with 
Amarillo that : (1) They are of practically the same 
population; (2) they are all in the same section of the 



Collection of Data 



69 



country; (3) the various items given in the table are on 
the whole rather close together for the different cities. 
Had he included some eastern, northern, or extreme 
southern cities of the same population, undoubtedly the 
variations in these items would have been much greater. 
Examples of Bad Sampling. Failure to pick out 
samples that really represent the whole group may lead 
to some very fallacious conclusions. For instance, 
Professor Bobbitt in his article on the cost of instruction 
in high schools, for purely illustrative purposes, compares 
results from various cities. ^ The population of the cities 
he uses varies greatly. Figures could not be obtained 
for all, but Table 5 gives enough to show the variations. 

Table 5. Variations in Population of the Cities Used by 
Professor Bobbitt in His Study of High School Costs 



City 


Population 
in 1910 


City 


Population 
in 1910 


Mishawaka, Ind. 

Elgin, 111 

Maple Lake, Minn. 
Granite City, 111. . 
East Chicago, 111. 
De Kalb, 111. . . . 
San Antonio, Tex. . 
Harvey, 111. . . . 
Waukegan, 111. . . 
South Bend, Ind. 
East Aurora, 111. . . 
Rockford, 111. . . 


11,885 
25,976 

9,903 

8,102 
96,614 

7,227 
16,069 
53,684 
29,807 2 
45,401 


Booneville, Mo. . . . 

Brazil, Ind. .... . 

Leavenworth, Kan. . 
Greensburg, Ind. . . 
Morgan Park, 111. . . 
Noblesville, Ind. . . 
Norfolk, Neb. . . . 

Washington, Mo. . . 
Bonner Springs, Kan. . 
Russell, Kan. . . . 

Junction City, Kan. 
Mt. Carroll, 111. . . . 


4,252 
9,340 
19,363 
5,420 
3,694 
5,073 
6,025 
3,670 

5,598 
1,759 



1 Bobbitt, J. F. : "Cost of Instruction in High Schools," School 
Review, 23 : 505-534. (Oct., 1915) 

2 Population for Aurora. There are two high schools, one in East 
Aurora and one in West Aurora. 



70 



School Statistics and Publicity 



But Professor Updegraff has shown that the cost of 
instruction is higher in large cities than in small ones. 
See Table 6. 

Table 6. Variations in Cost of Instruction and Supervision 
PER Capita (Population) in Cities of Different Sizes ^ 







Per capita 


Per capita 
expenses of 




Median 
city in 


Population 


expenses of 
teachers' 
salaries 


salaries and 
expenditures 

for 
supervisors 


Total 


Group I 


300,000 and over 


$50.98 


$1.18 


$52.16 


Group II 


100,000 to 200,000 


36.15 


3.26 


39.41 


Group III 


50,000 to 100,000 


36.93 


3.39 


40.32 


Group IV 


30,000 to 50,000 


29.25 


3.38 


32.63 



As to costs in schools in cities below 30,000, no such figures 
are known to the author. It is entirely probable that they 
are higher than the lowest figures cited here because of not 
having such full classes in high school. But the main 
point is that there are always great dangers in selecting 
samples that cannot reasonably be regarded as coming 
from the same class. Professor Bobbitt was, of course, 
perfectly aware of this difficulty and on page 506 of his 
article indicates that his tables are valuable as patterns 
of work mainly. 

Another example of the results of unwise selection 
of samples is furnished by Superintendent Spaulding's 
monograph on the cost of the Minneapolis schools.^ 

1 Adapted from Updegraff, Harlan : "A Study of Expenses in City- 
School Systems," Bulletin U. S. Bureau Education, 1912, No. 5, 
pp. 7, 86. (Computed on enrollment of pupils) 

2 Spaulding, F. E. : Financing the Minneapolis Schools. (Board 
of Education, Minneapolis, Sept., 1916) 



Collection of Data 71 

He has a chart on page 46 showing the expenditure per 
child for ordinary maintenance of the elementary schools. 
But in this chart he has included three southern cities, 
Louisville, Birmingham, and New Orleans. All three 
have in their population large negro elements which 
seldom pay taxes and add little wealth to the community 
because they are not able to do so. At the same time 
they have relatively large numbers of children to be 
provided with schooling. Consequently these southern 
cities have to take care of a large number of negro children 
with no corresponding increase in revenue. Therefore, 
they should not be placed in comparison with Minneapolis. 
Their presence in this table has the effect of making 
Minneapolis appear more extravagant in her expenditures 
for schools than would be the case if only northern cities 
were considered. 

Summary of Rules for Sampling. These may be 
given briefly: 

1. Be sure that the measures of any one item represent its usual 
state. 

2. Select samples absolutely at random. 

3. If there are groups of data, take the same percentage of samples 
from each group. 

4. Avoid using a questionnaire if possible. If it must be used, be 
careful to discount the results for classes liable to be selected by it. 

5. If only a small number of samples are obtainable, select them 
with unusual care. 

4. Blanks and. Tabulating 

Devices for Blanks. The actual collection of data 
should be made on some form for tabulating, called for 
convenience a '' blank." Rules for making blanks which 
hold good for many specific problems are extremely hard 
to lay down, but the following devices are often helpful : 



72 



School Statistics and Publicity 



a. Plan the blanks so as to get a maximum amount of 
information with a minimum of space. 

This reduces chances for error, because the more nearly a person 
can see all of the blank at one look, the greater mastery of it and 
of the relations of its parts he will have. But this is not to be inter- 
preted as advocating extreme condensation resulting in eye strain, or 
the elimination of all space for calculations. Nor does it mean that 
setting all the material on one sheet of paper is sufficient. A blank 
may be so long or so wide as to be hard to understand at one look. 

6. Use " double distribution " tables where possible. 

By these are meant " two-way " tables, — tables that classify data 
two ways, across the page on one classification, and down the page 
on another. They thus enable one to get the data from at least two 
separate tables condensed into a convenient form in one. Tables 7, 
8, and 9, with slight modifications, are taken from Snedden and 
Allen's School Reports and School Efficiency ^ to illustrate double dis- 
tribution tables. 

Table 7. Blank for Showing Number op Pupils Making 
Given Number op Days' Attendance 



First grade . 
Second grade 
Third grade 
Fourth grade 
Etc. . . . 



From 
19 days 



From 

20 to 

39 days 



From 

JfO to 

59 days 



From 

60 to 

79 days 



From 

80 to 

99 days 



Etc. 



Explanation: This table provides for a distribution of pupils' 

attendance in two ways: (1) by the grade; (2) by the number of 

days. Hence it is called a ''two-way" or "double distribution" 

table. 

1 Pp. 130, 132 



Collection of Data 



73 



Table 8. Blank for Showing Distribution of Pupils in 
Each Grade by Ages, Records of Ages Being Made 





First grade 


Second grade 


Totals 




Males 


Females 


Males 


Females 


Males 


Females 


From 5 yr. 
6 mo. to 6 














yr. 6 mo. 
From 6 yr. 
6 mo. to 7 














yr. 6 mo. 
Etc. . . . 















Note : This is a very convenient table, since it gives a distribution 
of pupils both by ages, and by grade and sex. 

c. Strive to have all the printing on the blank so that it 
can he read from one position while filling in the data. 

Table 7 was originally printed like Table 9. 

Table 9. Blank for Showing Number of Pupils Making 
Given Number of Days' Attendance 







CO 


CO 


CO 


CO 






CO 


S^ 


^ 


STi 


>5 






?r5 


;3 


e 


!3 


S 






e 


^ 


-^ 


^ 


^3 






'XS 


05 


Oi 


Oi 


Oi 






Oi 


<50 


^ 


t^ 


Oi 






•»-s 


O 


o 


o 


o 






O 


"*-^ 


'*^ 


-»^ 


-K) 






-»o 


o 


o 


O 


O 






H|(N 


(^ 


-^ 


^ 


oo 






g 


g 


s 


g 


g 






Q 


o- 


o 


o 


o 


«:> 




^ 


^ 




V. 








fcn 


c^ 


^ 


!*H 


t^ 


uq 


First grade . . . 














Second grade . . 














Etc 















74 



School Statistics and Publicity 



Note: This is exactly the way the blank should not be made up. 
The proper form is given on page 72. With care, the space usually 
can all be utilized either way. If all the titles and headings can be 
read from one position, that is, without having to turn the paper 
around, the chances for error are much reduced. 

d. Put full or plainly abbreviated title, not merely an 
arbitrary symbol or a number, above each column, if any 
one besides the superintendent is to use the blank. 



4. NATIONAL SCHOOL RECORD SYSTEM 

TRANSFER CARD 



To be filled out for the Attendance Offi- 
cer in case of transfer to any other 
school, either in or outside city. . . . 



1. Last name 



2. First name and initial 



6. Name of parents or 
guardian 



8. Residence before 
discharge 



7. Occupation of parent or 9. Date of discharge 
guardian 



8. New residence (or name 
of private or parochial school 
if pupil is transferred to one) 



10. Age when discharged 
Years Months 



d. 

Grade 


e. 

Room 


f. 

Days Present 


Health 


h. 

Conduct 


i. 

Scholarship 


Date of last attendance 


School 


District 


Teacher 


Principal 



Remarks on other side 



THE SHAW-WALKER CO., MUSKEGON FORM NO 4 



Some superintendents get out blanks of the kind condemned here. 
But it is very doubtful if such blanks will ever secure accuracy from 
anyone except the person who thought out the key (even he may 
forget it if it is at all intricate) and the small percentage of teachers 
who may be called "hyper-patient." Teachers dread such blanks 
for precisely the same reason that most readers dread having to refer 
constantly to footnotes in something they are trying to read. 



Collection of Data 75 

If the data are to be copied or read to some one else, a key num- 
ber should appear in each column immediately under, or in front of, 
the title. The reader or copyist can catch this key number much 
more quickly and easily than he can the heading. The key number 
is also preferable for calling or reading off to some one else. 

A good example of the use of key numbers and letters with titles 
is found in the National System of Uniform Records, the transfer 
card of which is reproduced on page 74. 

e. If the same item appears in different blanks of a 
series, always give it the same key number and the same 
relative position to closely connected items. 

The report of the Committee on Uniform Records and Reports, 
previously referred to, recognized this principle. The order of the 
various items appearing on the pupil's report card is the same as 
that appearing on the lower half of the attendance and scholar- 
ship record kept in the teacher's loose-leaf register, which is later 
filed in the principal's office. Other cards in the system which 
contain these items use the same key letters and relative position. 

/. If data are to be summarized from figures furnished by 
many persons, make the summary with as little copying as 
possible. 

Copying always tends to error, and the checking necessary to avoid 
such errors is very laborious and time-consuming. The special econo- 
mies here are things like the following: (1) In a building, the data 
for a report on the table given on page 72 may be secured by hav- 
ing each teacher call at the principal's office and put her data on 
the proper horizontal line. The completed blank will then be the 
desired summary. (2) If the data are to come from different build- 
ings, each teacher can put hers on the proper line on a blank and 
send it to headquarters. There such blanks can be gathered, the 
line clipped off each blank (keeping necessary identification marks of 
course), and pasted on a sheet of paper, thus forming the summary 
at once. In very extensive or frequent tabulations of this sort, 
blanks with perforations where the lines are to be cut apart may be of 
service. Perforated blanks are, however, very expensive and usually 



76 



School Statistics and Publicity 



unnecessary. For most purposes, cutting and pasting, or copying 
and checking, will be sufficient. 

g. Figure or summarize on the blank itself if time can be 
saved thereby, as is often the case. 



Sabjact P^WSt'cS 



Class 



Teacbe 



r Li^S 



Co Vvi 



Ir 



Sex 



Al 



Tears Erperience, before 1913-14_ 
Head of Department? 



:^5'-l05 ; 90.94 : 80-89 : 65-79 : 150-701 ; 7SA : Below 65 : Total 



Boys : 


/ = 


\ 


-ffH- '- 
II 


t 


, / '■ 




; ; 






1^ 


1 


1 

® 

1 — ^ 




Girls J / 




/// 


. // 




; 


1 • , • 


; / 

;<2) 


: ^ 






o 






= / 


• 7 


■ If 


(D 


: / 


Id \31 



Data for Grading System - Hume -Fogg High Sobool 
Grades for 1913-14. 

FiQ. 1. — Example of Scoring and Figuring on the Blank Used for Collect- 
ing Data. 



Collection of Data 77 

In summarizing the material collected as advised in section /, the 
additions can be made in ink (often red ink) on the bottom of the 
sheet where all the data have been collected. A good illustration 
of this point is afforded by a blank which was used by a class of 
graduate students making under the writer's direction a study of 
the marks given in the Hume-Fogg High School in Nashville.^ (See 
Figure 1.) 

In Figure .1 original scores are indicated by the marks above the 
line. The plain figures below each line are the numerical equiva- 
lents of the scores. The figures in rings represent the approximate 
percentage values of the numbers just above them. On the original 
blank such percentages could be indicated by red ink without using 
rings. All percentages are calculated here to show the method. But 
in this particular problem, because of the small number of cases in 
each group, such percentages have little value. See pages 13, 63. 

h. In case only one copy of a blank can be made, use 
cross-section paper, or ruled blank book. 

Cross-section paper saves an enormous amount of time in ruling 
paper and avoids eyestrain. Such paper may be purchased from a 
good scientific supply or drawing company, in sheets of various sizes. 
That particular kind characterized by heavy lines every five or ten 
small squares is the better kind. The heavy lines serve as columns, 
while the small squares help to keep figures under each other. 

A ruled blank book or account book, especially the kind extending 
over two adjacent pages, is similarly useful because of the vertical 
columns and horizontal lines. 

i. Put at the top of the blank the item which determines 
the place of the blank in a series or which identifies it. 

The observance of this caution will make for ease in handling the 
data. For example, in the case of cards that are to be filed alphabet- 
ically according to the name of the student, this name should appear 
at the top of the card where it will be the first thing the eye of the 
one going through the records will light upon. Yet many school men 

1 The results of this investigation appeared in Educational Ad- 
ministration and Supervision, 1 : 648 (Dec, 1915), from which article 
this illustration is adapted. 



78 School Statistics and Publicity 

get out blanks with the name of the school, the name of the card, the 
name of the town, and other insignificant items at the top, which is 
the choice position for identifying a blank quickly, especially if it is 
ever filed. Such material is very seldom handled except by persons 
who use the blanks more or less constantly. For these, this material 
may as well come at the bottom of the card and often even in fine 
print. Even the transfer card of the National School Record System 
shown on page 74 could be improved upon in this respect. The 
card is to be filed alphabetically by the name of the pupil. This 
name should come on the top line, and the title of the card with the 
caution to the attendance officer should be placed at the bottom. 

j. Always use blanks cut to standard sizes. 

This is essential for easy handling of data, filing for ready refer- 
ence, and storage for future use. Folders and filing cases can easily 
be procured for standard sizes, and the printer or supply house can 
furnish paper and cards for these sizes at much lower prices than for 
odd sizes. The standard sizes are 3x5, 4x6, 5x8, and 8|xll inches. 

Examples of Good Blanks. The busy superintendent 
may often save much time by getting good blanks used 
in other systems and quickly modifying them for his own 
purposes. Such blanks may be found in : 

1. Snedden and Allen : School Reports and School Efficiency. 
(Macmillan) 

This discusses the good and bad points in actual blanks from 
city school systems, on such topics as the school plant, expend- 
itures, the census, attendance, attendance and ages, promotions, 
summaries, etc. 

2. Report of the National Education Association Committee on Uni^ 
form Records and Reports, published as Bulletin U. S. Bureau Educa- 
tion, 1912, No. 3. 

It gives model blanks for the following in elementary schools : 

1. Principal's term report : (a) enrollment ; (6) promotions, 
non-promotions, by grades; (c) distribution of withdrawals 



Collection of Data 79 

by ages and causes ; (d) distribution of attendance ; (e) grad- 
uates by years in schools; (/) non-promotions by grades and 
causes ; (g) failures by studies and grades ; (h) distribution of 
leavings and withdrawals by ages and grades ; (i) ages of grad- 
uates; (j) enrollment and attendance; (k) distribution of 
whole-time teachers. 

2. Teacher's term report: (a) enrollment by divisions; 
(6) non-promotions by grades and classes ; (c) failures by grades 
and studies ; (d) enrollment and attendance ; (e) distribution 
of enrollment by ages ; (/) distribution of withdrawals by ages 
and causes ; (g) distribution of leavings by ages ; (h) beginners 
by training; (i) beginners by ages. 

There is also a set of blanks for high schools similar to these. 

3. Strayer and Thorndike: Educational Administration. (Mac- 
millan) 

This contains extracts from statistical researches on educa- 
tion made at Teachers College, Columbia University. It con- 
tains some of the blanks recommended by the National Educa- 
tion Association Committee as well as very valuable table 
forms on other things. 

4. Rugg, H. O. : Statistical Methods Applied to Education, pages 
28-87. (Houghton Mifflin) 

5. Certain Surveys. 

The Butte Survey, for example, contains good suggestive 
blanks on these phases : census, attendance, absences, educa- 
tion and experience of teachers, enrollment, promotion and 
failures, size of classes, ages of children at beginning of first 
semester, receipts and expenditures. 

One Blank versus Card Index. In many pieces of 
statistical work, one must early face the question as to 
whether it is better to put all the data on one large 
blank, allowing one line to each case or group, or to enter 
each case or group on a separate card. 

One of the best arguments for the latter method is that 
the data in the columns cannot show their full meaning, 



80 School Statistics and Publicity 

as a rule, unless arranged in order from lowest to highest. 
If the data are all on one large sheet, the classes may come 
in order for the data in the first column, but this order 
will not be the same in any one of the other columns, 
which may be several in number. If these columns are 
to be accurately studied, the data in them must be 
copied off and rearranged, with many chances for error. 
On the other hand, if no large table is made, the cards 
may be taken up in order easily for one column. The 
data may then be copied off for a table on this one item. 
Next, the cards may be rearranged for column two and 
the corresponding table made. This process is much 
easier mentally and more accurate than the first one 
mentioned above. It is the procedure used in the Courtis 
tests, where each child has a folder containing his own 
work. These folders are taken up and sorted into piles 
of differing achievement for attempts in addition, and 
from these a distribution table is made ; they are re- 
distributed for attempts in subtraction, and a second 
distribution table is made, this time for subtraction ; the 
process is repeated for multiplication and division. Then 
the start is made on ''rights," and the four sortings take 
place on this basis and the four resulting tables are 
copied off. 

To summarize, the advantages of each plan are : 

Large Sheet Card Index Plan 
The data are not easily lost. Less chance for error in copy- 
It is impossible to lose a part, ing or rearranging data. 

as can easily happen in the use Easier mentally. 

of cards. 

All data are before the eye at Any parts of the data may be 

one time, and thus a better pre- separated from the remainder at 

liminary grasp of the situation any time, 

may be had. 



Collection of Data 



81 



Less work in compiling at The data may be easily shifted 

first. to any order. 

At the beginning data may be Pupils can handle data better, 

copied in any order. Often it will be advisable to give 

portions to different workers. 

The disadvantages of these two plans are largely the 
converse of the advantages, but they may be emphasized 
through the statement of them: 



Card Index Plan 
If card is lost, it cannot be re- 
placed, nor can the loss be easily 
discovered. 

A card may stick to another 
and be easily overlooked. 

Cards for easy handling must 
ordinarily be kept in alphabetical 
order. Hence if shifted for pur- 
poses of making tables, etc., 
must be returned to regular order. 

Much mechanical work is neces- 
sary in examining data. 



Large Sheet 

Data from large buildings or 
groups not easily brought to- 
gether on one sheet. 

Additional data (often come 
late) are hard to insert. 

Data in all columns not ar- 
ranged from highest to lowest 
nor can they be so arranged with- 
out recopying. 

More chances for error in re- 
copying and rearranging. 

Will stand less wear and tear. 

Averages and summaries are 
hard to make unless data are ar- 
ranged in order from the start. 
Often this cannot be determined 
when data are copied on large 
sheet. 

On any given problem, a little forethought or practice 
on sample data will usually indicate which is the better 
plan. However, it is very difficult to be certain in this 
matter until the whole statistical procedure has been 
thought through. 

EXERCISE 

For your special problem : 
(a) Decide just where you would use a card index and just where 
one large blank, giving full reasons. 



82 School Statistics and Publicity 

(b) List the blanks, with very definite titles, that you now think 

you would need. 

(c) Draw off in complete detail, accurate to size of paper, at 

least one of these blanks. 

5. Miscellaneous Economies in Collecting Data 

a. Cross-section paper for scoring. Many times it is necessary to 
keep track of the number of cases, the data for which may come from 
widely separated places. Keeping track mentally is very difficult 
and inaccurate. It is far better to adopt some method of putting 
down a mark for each case as soon as it is located, and later count 
the marks. The cross-section paper device is one for such marking. 



































































































































































— 
















































1 


1 






1 


1 


1 


1 


1 


1 












< 










1 


1 






1 


1 


1 


1 
















)0 


-1 


0( 


) 


1 


1 






1 
































\ 


1 






1 
































1 


I 






1 






































































































e 


10 


"~ 


8J 


) 

















































































































Fig. 2. — Scoring on Cross-Section Paper. 

Suppose that a study is being made of the marks given by various 
teachers to pupils. A sheet, column, or whatever space is desired, 
may be assigned to each teacher. One horizontal row of large squares 
may be labeled 90-100 or A, according to the grading system ; another 
80-89 or B, etc. Each mark between 90 and 100 as soon as located 
would be represented by a check mark in one of the little squares in 



Collection of Data 83 

the row of large squares representing grades 90-100, under the column 
for the proper teacher or subject. For such work, always begin at 
the left upper corner of the first big square on the left and fill in reg- 
ularly to the right, row by row. Suppose, when the work has been 
finished, that the 90-100 column looks like the illustration given. 
One glance at the squares in Figure 2 shows that there are 33 grades 
between 90 and 100 given by the teacher during the time under in- 
vestigation. 

b. Scoring by fives on plain paper. This is the old device used in 
counting the votes in elections, etc. The first four cases of every five 

WW 

m 

THi 

^^ or M m w tm mi rtu in 

m 



Ill 



Fig. 3. — Scoring on Plain Paper. 



are given a straight perpendicular mark each ; the fifth is made hori- 
zontally, tying the other four, as in Figure 3. 

This makes 33 cases, as in the preceding method of scoring. For 
most people the counting is much easier if the groups of five are under 
each other and not in a horizontal line. 

The author also has made much use of the scheme of putting down 
a mark for every case in the group as best he could for speed, on any 




Fig. 4. — Scoring on Plain Paper. 

paper handy. Then rings are drawn around the marks with five in 
each ring. It is easy to count the rings. The chief merit of this 



84 



School Statistics and Publicity 



scheme for a single worker is that the eye does not have to be shifted 
quickly and accurately, with the resulting strain. Putting down the 
dots roughly requires no eyestrain. Putting the rings around the 
dots is not hard, as the eye is kept focused on the particular part of 
the paper. Thus, the same 33 cases might be scored as in Figure 4. 
c. Checking on blanks. Where possible, tabulate by check marks 
in appropriate column on a blank, as in the United States Govern- 
ment stock device. Thus, a superintendent could study physical 
defects of school children with a blank like Table 10. 

Table 10. Blank Illustrating Method of Checking in 

Entering Data 







Physical Defects 










Sight 


Hearing 








Pupil No. 






Ade- 
noids 


Tubercu- 
losis 


Efr 














Right 


Left 


Right 


Left 








1 






X 










2 




X 












3 
















4 
















5 


X 


X 






X 






6 








X 








7 






X 


' 









d. Some entry for each case. In some instances accuracy is in- 
creased by the device of making some sort of entry for each case. It 
is very easy to omit a few cases from a large number if entries are 
made only where data actually exist. But if numbers are put down 
for actual data, zeros if it is known that nothing is done, and "n. d." 
(for "no data") where it has been impossible to secure data, the 
results will be likely to be much more accurate. The necessity of 
having to account for each case in a positive form, reduces the chances 
for omission, a fact long ago discovered by insurance companies, 
which require their agents to make some report on each item on a 
blank. 



Collection of Data 



85 



e. Rider strip for printed reports. Often it is necessary for the 
superintendent to get figures from two or three widely separated 
columns in a table of fine print. There is great eyestrain and many 
chances for inaccuracy in trying to copy them off directly, especially 
where the tables run over two pages, as do those in the government 
bulletins. The best way to avoid this is to cut out a strip of paper 
which may be placed over the table so that the desired figures will 
stand out clearly and quickly in the angles of the paper. Table 11 
indicates how to work the scheme if the superintendent wishes to 
compare his system with the cities in the table in the matter of ex- 
penditures under the headings, Board of Education and Business 
Offices, Superintendent's Office, and Other Supplies. Government 
offices use specially constructed rulers for the same purpose. 

Table 11. Illustration of Use of Ruler Strip Device on 
A Page, in the Report of the U. S. Commissioner of 

Education 



Cities 


B'd of Ed. 

S^Bus. 

Offices. 


Supf's 
OffTce. 


Sal. 
^Exp 

of 
super- 
visors. 


Sal 

of 
?rm. 


Sal.of 
Teachers 


Text 
Bool<5 


Other 
Supplies 


ALABAMA 


$3631 
128163 


$10750 
43801 










$5967 
J05600 


Birmingham 

CALIFORNIA 

L,osAn3etes 


J 


1 1 

1 1 




1 .. ^ 














Another very convenient form of ruler strip is made with two 
strips of zinc adhesive or bicycle tape and some thin, tough paper. 
The strips of adhesive are pasted across the paper, but a little apart. 
Then the top strip is cut across, the cuts corresponding to the lines in 
the table to be used. Then some of the resulting flaps may be pulled 
back and pressed back to show the figures in the desired columns, 
as in Figure 5. 



86 



School Statistics and Publicity 



The advantages of this form are that it can be used for various 
combinations of columns, and that it is very durable. 

/. One operation at a time. The idea is to carry the same step 
or operation all the way through without stopping to do something 
else. That is, if one is preparing a table from data found in different 
tables, he should copy all the data from each table in turn, and not 
skip from one to the other. Or if one has several groups of data to 



Papi 



Fig. 5. 



/ap« 


- 


43801 


I 1 

1 1 

1 1 


105600 










-——-"— 





















Illustration of Use of Adhesive Tape Ruler Strip on Figures of 
Table 11. 



rearrange, he should finish each group before going to the next. This 
gives the practice effect for that operation, insures greater accuracy, 
and is much easier mentally. 

g. Using high school students to gather data. For gathering many 
of the data, high school students can do as well as any one else. A 
class can collect, classify, and check a great deal. Of course the 
question at once arises as to whether this is a legitimate use of the time 
of high school students, who supposedly go to school for their own 
benefit and not to help work out school statistics. But many of these 
are later to do clerical work of various sorts. Much of the statistical 
work in school affords the finest sort of clerical practice for such stu- 
dents. Their welfare will be properly cared for if the work is done 
under the careful supervision of the superintendent. He will natu- 
rally discover a great deal of valuable knowledge about the vocational 
aptitudes of his individual students for satisfactorily doing clerical 
work or for directing others in such work. Accordingly it may be 
considered thoroughly sound preliminary training for them, as Pro- 
fessor Bobbitt has pointed out.^ '^ 

But there is more direct evidence of the value of such work than 
any such theoretical statement. In 1915 one of the author's graduate 
students, Mr. S. J. Phelps, later professor of secondary education at 
the University of Vermont, directed the work of the students of the 

1 San Antonio Survey, pp. 32, 33. 



Collection of Data 87 

Gallatin, Tennessee, High School in making a study of four problems 
connected with their school system.^ These problems were : 

1. Cost of maintenance of the system. 

2. Age-grade distribution of the pupils. 

3. Variations in marks given by high school teachers. 

4. Study of the lighting facilities in each room. 

Clear and concise instructions were placed by Professor Phelps in the 
hands of the various teachers engaged in the work. The answers of 
the students were carefully checked over in class and then submitted 
to the inspection of a graduate class in school administration at George 
Peabody College for Teachers. Most of these men had had much 
experience in administrative work and all were specializing in survey 
work. They unanimously agreed that the work was surprisingly 
accurate. 

In answer to the question : Is not such work an exploitation of the 
students? Professor Phelps answers: 

"All are agreed that in mathematics, especially, it is necessary 
for a high school student to do a large amount of drill work. Now 
in doing this, which is the more profitable and practical for a high 
school student who is studying, for instance, eleventh grade civics 
or arithmetic, to do as outside work : To study the costs of oper- 
ating his own school system, compared to similar costs in other 
towns, or to find out the number of days it will take A, B, and C 
working together to do a piece of work which A can do in three 
days, B in four days, and C in five days? 

"How would a study of costs in his school compare with a paper 
which he might, after much delving, prepare on the source of some 
abstract principle of governmental costs? Isn't this a place where 
the much-talked-of subject, Community Civics, could get some 
practical problems ? 

"Which seems more practical and profitable for a class in algebra 
studying the graph, — to make a graph of these same costs, or to 
graph the profile of a river bed, or perhaps an extract from the 
table of American Mortality Experience? 

"In which would a student in percentage be expected to show 
. the more interest, — in a study of the percentage distribution of 

1 Phelps, S. J. : Master's thesis at George Peabody College for 
Teachers, 1915, on file in library. 



88 School Statistics and Publicity 

the marks given by his own teachers, among which he has a mark, 
or in studying the percentage composition of some compound, 
perhaps a fertilizer, which he has never seen and in which he cannot 
be expected to show a passing interest or curiosity? 

"Would another student in practical measurements get more 
from computing the surface and volume of the earth, or from find- 
ing how many spheres of a certain diameter could be placed in a 
cylindrical cup of certain dimensions, than he would get from 
studying the ratio of lighting space to floor space and air space per 
pupil in the same room?" 

From such suggestive questions as these it may be surmised that 
work of this sort, instead of "exploiting" the high school student, 
would be of great practical benefit to him. 

A report of similar work from Wisconsin is as follows : 

HIGH SCHOOL CLASSES 
GRAPH CONDITIONS 

Over-Age and Failures Studied 



In connection with the algebra work 
in the Frederic High School, graphic rep- 
resentations in colors are made showing 
conditions in the number of students re- 
tarded, and other school problems. One 
percentage graph compares the percentage 
of students over age, showing a distinct 
decrease in retardation during the past 
four years. Other graphs include mate- 
rial on students dropped, failed, and pro- 
moted in various subjects. Teachers are 
also compared with respect to the number 
of students failed by each of them. 

Incidentally, such work as this gives the 
algebra pupil practical work to do, illumi- 
nates the general subject of graphic analy- 
sis, and makes mathematics interesting.^ 

1 Wisconsin State Department of Education : Educational News 
Bulletin, Jan. 1, 1917. 



Collection of Data 89 

EXERCISE 

Which of the miscellaneous economies would be applicable to your 
problem, and just how would you use them? 

REFERENCES FOR SUPPLEMENTARY READING 

King, W. I. Elements of Statistical Method, Chapters IV-IX. 
Report of the Committee on Uniform Records and Reports. U. S. 

Bureau of Education Bulletin, 1912, No. 3. 
Rugg, H. 0. Statistical Methods Applied to Education, Chapters II, 

III. 
Thorndike, E. L. Mental and Social Measurements, Chapter II. 



CHAPTER III 

TECHNICAL METHODS NEEDED IN SCHOOL 

STATISTICS 

I. USUAL VIEWS 

So far we have considered only statistical matters 
that are plain to any experienced school man. With the 
suggestions previously given, such a man could success- 
fully collect statistical data on most of his school prob- 
lems. But the working up of the data and the proper 
interpretation of them would be altogether different 
and much more difficult. He would at once face 
such questions as these: Does the superintendent need 
any technical knowledge of statistics? Can he, without 
such special knowledge, analyze his data and get the 
really significant things out of them? Or, without such 
knowledge, can he present these results effectively to the 
public ? 

The Conservative's View. Attempts to answer these 
questions have brought forth much nonsense and fruit- 
less effort. For example, one group of school men take 
the position that a superintendent needs no special knowl- 
edge of statistics, simply because there is in their opinion 
no virtue in statistics. They quote the old statement of 
Bagehot : " There are three kinds of lies — lies, damned 
lies, and statistics." Or they reiterate : " Figures don't 
lie, but liars do figure." Or else they would at least 

90 



Technical Methods in School Statistics 91 

agree with the author of a recent article that '' whatever 
the causes, the fact is that any one who presents his 
arguments in the form of tables, and his conclusions in 
dogmatic statements presumably based on the tables, 
is sure to convince nine tenths of his readers." ^ Con- 
sequently, superintendents having this viewpoint see no 
need of any special knowledge of statistics. They think 
that any effort to do anything unusual with statistical 
data is simply a waste of time. The absurdity of such an 
opinion should be evident to any one who has even glanced 
through the preceding chapters of this book. Professor 
King states the whole point very strikingly thus : ''To 
attempt to handle statistics properly without a knowledge 
of statistical method is only a little less absurd, though 
vastly more common, than to attempt to build a great 
steel bridge without a knowledge of trigonometry." ^ . 

The Specialist's View. Another group, composed of 
scientific educators and statistical experts, advocate 
either a very thorough course in statistical method or 
none whatever. They quote Pope on " A little learning 
is a dangerous thing," etc. Or they say that giving a 
school man only a little, or very superficial, knowledge of 
statistics is like putting a razor in the hands of a baby. 
Or they compare the results of such a procedure with those 
ensuing when very delicate and expensive machinery is put 
in the hands of a novice. Such machinery, which would 
produce wonders if run by a competent man, is, of course, 
soon ruined by a bungler, and the result produced is very 
inferior or altogether lacking. Even Professor King says : 
" The science of statistics, then, is a most useful servant, 

1 "Lies, Damned Lies and Statistics," Unpopular Review, 1915^ 
Vol. II, 352-353 

2 King, W. I. : Elements of Statistical Method, pp. 37-38 



92 School Statistics and Publicity 

but only of great value to those who understand its 
proper use." ^ 

The Golden Mean. The truth probably lies between 
these extremes. Even Professor Thorndike, the pioneer 
in the application of modern statistical method to edu- 
cational problems, favors this moderate view when he 
says : '' There is, happily, nothing in the great principles 
of modern statistical theory but refined common sense, 
and little in technique resulting from them that general 
intelligence cannot readily master." ^ Of course, he 
later says that mathematical gifts and training will be 
very useful to students of quantitative mental science, 
but such things are not absolutely necessary for learning 
the elements of statistical method. 

Observations in other fields also support this " golden 
mean " view. Often the successful politician, minister, 
or business man has better practical ways of controlling 
people and reading human nature than has the expert in 
the psychology of such matters. It is not an uncommon 
thing for an experienced public speaker to influence an 
audience more than a teacher of public speaking could 
hope to do. We are only saying in other words that 
common sense and first-hand experience in reading and 
controlling human minds are as powerful factors in in- 
fluencing the public as expert knowledge in the mechanics 
of any art having the same end in view. Is it not reason- 
able, then, to expect the experienced superintendent 
with a small amount of statistical theory to outdistance, 
in practical statistics with the public, the best-trained 
experts in statistical theory only? 

1 King: oj)- cit., p. 33 

^ Thorndike, E. L. : Mental and Social Measurements, p. 2 



Technical Methods in School Statistics 93 

II. STATISTICAL KNOWLEDGE NEEDED FOR SCHOOL 

SURVEYS 

But another phase of this question arises. The modern 
superintendent must be able to read and understand 
school surveys and apply the statistical methods used to 
his own school problems. How much statistical work 
does he need for this sort of thing? 

A study of practically every school survey thus far 
published shows that to meet this requirement the 
superintendent should, in addition to things previously 
mentioned, know : 

1. The meaning of these terms : median, average, quartile, range, 
central tendency, variability, overlapping, and coefficient of varia- 
bility. 

2. The methods of computing medians, quartiles, averages, and 
variability so that any fallacies or mistakes arising from poor or false 
methods may be detected. 

3. How to read and understand tables and graphs. 

4. The principles underlying the theory of good and bad units, and 
how the unit in the particular study was derived. 

5. The principles underlying the construction of graphs, so that 
he will not be misled by a badly constructed graph. 

6. How scales are derived, so that he will not be misled by the 
interpretations made from data measured on these scales. 

However, the citation of survey material as evidence 
of the relatively small amount of statistical knowledge 
needed by the superintendent is open to at least one 
criticism, that the surveys are written for the public, and 
to be effective must contain little or no technical material. 
But the proper translation of such facts into popular 
language presupposes accurate statistical work. And 
few superintendents could hope to do more than to give 
the best possible statistical treatment to such problems 
as are so treated in all our best school surveys to date. 



94 School Statistics and Publicity 

III. STATISTICAL KNOWLEDGE NEEDED FOR READING 
EDUCATIONAL INVESTIGATIONS 

Moreover, for publicity work a good superintendent 
needs to keep up with recent investigations in education 
and psychology, as carried on by investigators in the 
various schools of education and by educational founda- 
tions. Some of the most valuable of these are not written 
especially for laymen and require some knowledge of 
statistical terms and methods to be understood at all. 
Good abstracts or news items of such studies often employ 
these terms. The superintendent needs to understand 
the terms and general processes used, but he need not 
know how to calculate or employ these terms accurately 
himself. It would be better if he could so use them, but 
it is not absolutely necessary. The situation is similar 
to that in reading for all of us. We must know how to 
employ, use, and spell correctly certain words. We need 
to be able to read with sufficient understanding a much 
larger list of words. But it is not necessary for us to 
know how to spell these latter words, or how to use 
them correctly in our speech or writing. 

The superintendent needs to understand the meaning, 
for reading purposes, of at least the following : 

1. Such terms as : average deviation, standard deviation, mode, 
probable error, inter-percentile range, correlation, skewness, dis- 
persion. 

2. The reliability of the various measures of variability. 

3. The effect of different methods of grouping data, on the con- 
clusions reached. 

4. Some of the common methods of making allowance for the un- 
reliability of data. 



Technical Methods in School Statistics 95 

IV. ILLUSTRATION OF VALUE OF STATISTICAL METHOD 
TO THE SUPERINTENDENT 

The value of statistical method and presentation to the 
superintendent may be most clearly presented by a con- 
crete illustration. Suppose a superintendent wishes to 
know whether his high school classes are too large or 
too small for good work. He may take as his standards 
the pronouncements of colleges or universities, or the 
actual classes found in high schools that he or some 
competent person rates as good ones. 

He would, of course, get the enrollments of all the 
classes in his own high school, say fifty or more. He 
would get similar figures for twenty other high schools, 
or more if possible, say for at least a thousand classes. 
Then he would be confronted with the problem of handling 
this enormous mass of data so as to bring any clear idea 
out of it. His data would cover pages and pages. He 
would have an unwieldy mass of facts that needed sim- 
plifying. Without proper treatment, his ideas of the 
whole would be '' decidedly vague and indefinite." He 
would need some procedure that would enable him to 
" give clear-cut form to this hazy conception " and to 
" set objects in their proper perspectives and relation- 
ship." ' 

Now if he knew the elements of statistical method, he 
could very shortly summarize his data on a half page, as 
Professor Bobbitt does in the School Review for October, 
1915.2 In this article, to which we have previously 
referred, Professor Bobbitt is making studies of the cost 

1 King, W. I. : Elements of Statistical Method, p. 28 

2 Bobbitt, J. F. : " High School Costs," School Review, 23 : 505- 
534 



96 



School Statistics and Publicity 



of instruction per one thousand student hours in a number 
of high schools. First, each study, as English, mathe- 
matics, etc., is worked out separately as shown in Table 
1, on page 18 of this book. Table 12 is the summarizing 
table. 

Table 12. Bobbitt Table Showing Sizes of High School 
Classes by Subjects 



1 



Median 
No. Pupils 


" Zone of S a 




Pupils 


58 


42-88 


32 


28-55 


22 


20-24 


21 


18-24 


21 


17-23 


20 


16-22 


19 


18-25 


19 


15-23 


18 


14-24 


17 


15-20 


17 


14-19 


17 


13-23 


15 


10-21 


14 


12-18 



Music 

Physical Training . . 

English 

Mathematics . . . . 
History ...... 

Science 

Agriculture . . . . 
Commercial . . . . 

Drawing 

Modern Languages . . 

Latin 

Household Occupations 
Normal Training . . 
Shopwork 



To any one who understood the simplest things about 
statistics, this table would at a glance disclose such facts 
as these: In music, half the schools have more than 58 
pupils in a class, and half have less; half of them have 
between 42 and 88 pupils; a fourth have less than 42 
pupils; a fourth have more than 88 pupils. Similar 
statements would hold for the other subjects down the 
table. The table would also disclose that the " average 
classes " are in agriculture and commercial subjects. 



I 



Technical Methods in School Statistics 97 

with 19 pupils each as a rule. Half the classes in other 
subjects have more than this number and half of them 
have less. The table would show which had more and 
which less, ranging from music with 58 down to shopwork 
with 14. A little more inspection would disclose which 
classes ranged more in their variations from the ''typical" 
or '' average " class in that subject. All these things, if 
told in words, would occupy pages and pages of description 
that would be about as clear and interesting as a real- 
estate deed to the lot on which the superintendent's home 
stood. 

The Bobbitt table,^ of course, is infinitely clearer and 
more forcible than the great masses of data or the long 
and tedious description could ever be. It has indeed 
fulfilled " one of the prime objects of statistics." This, 
according to Professor King,^ is " to give us a bird's-eye 
view of a large mass of facts, to simplify this extensive 
and complex array of isolated instances and reduce it to a 
form which will be comprehensible to the ordinary mind." 

V. STATISTICAL METHOD AS A FORM OF EXPRESSION 

Finally, the superintendent needs statistical method 
just as he does any other method of effective presentation 
or expression. A good description of scenery, of an 
object, a face, etc., always proceeds by giving first a 
bird's-eye view, or very brief comprehensive sketch, 
called usually the " fundamental image," or in exposition, 
" the topic sentence." Then the details are later filled 
in. The success of the description or explanations depends 
upon the clearness, brevity, and vividness of the funda- 

1 The writer uses this name because such tables appear to have 
been first used by Professor J. F. Bobbitt of the University c:f Chicago. 

2 King : op. cit., p. 22 



98 School Statistics and Publicity 

mental image, and then upon the extent to which approxi- 
mately all the important details place themselves clearly 
under it. Of course, the whole process must not be so 
mechanical and obvious as to disgust the reader. But 
without the fundamental image and this procedure, the 
reader would soon be utterly lost in the details, or he 
would get so few of them in mind, or in such a disorganized 
manner, that he would get no clear idea of the whole. 
And he would not waste any more time in trying to do so. 
In the same way statistical method forces a mass of 
numerical data into a form which describes the whole 
by giving a good fundamental image or picture. At the 
same time it leaves all the data so grouped and classified 
that significant points stand out, but both points and 
minor details of any consequence may all be easily located 
under the fundamental image. 

If suitable graphic presentations of the statistical results 
are made, they will have the same advantage that a 
line drawing has over a photograph. The superintendent 
without a knowledge of statistics might give in words 
only a picture that would correspond to a photograph 
taken without proper focus. This, of course, gives all 
the details, but blurred. A suitable graph would make 
the essential facts of the whole and the essential details 
stand out clearly. It has been found repeatedly by text- 
book writers for beginners that line drawings emphasizing 
the essential elements in apparatus, pictures, etc., are 
preferable to actual photographs of the objects, because 
the photographs give too many details and so obscure 
the big things. The superintendent with no knowledge of 
statistical method could, under the most favorable cir- 
cumstances, give his reader only a sort of blurred photo- 
graph of his ideas. And, indeed, he would probably have 



Technical Methods in School Statistics 99 

only a blurred picture of them in his own mind. With 
statistical method he could have in mind a sharp line 
drawing and give his readers the same kind of picture. 
The essentials of this method that are of value to him 
will be taken up in the next chapter. 

EXERCISES 

1. Which of the statistical terms mentioned on page 93 do you 
think you understand fully? 

2. Which of the statistical processes given on the same page do 
you think you know how to do ? 

Note : The best way to know whether you understand a term fully 
is to see whether you can quickly write a clear explanation of it. 
Similarly, to know whether you can tell how to do a process, see if 
you can quickly write directions for doing it. 

3. Which of the statistical terms mentioned on page 94 have you 
come across in your reading, and just what do they mean to you at 
present ? 

REFERENCES FOR SUPPLEMENTARY READING 

King, W. I. Elements of Statistical Method, Chapters I-III. 
Rugg, H. 0. Statistical Methods Applied to Education, Chapter I. 
Thorndike, E. L. Mental and Social Measurements, Introduction 
and pages 36-41. 



I 

I 



CHAPTER IV 

SCALES, DISTRIBUTION TABLES, AND SURFACES 

OF FREQUENCY 

Thus far we have discussed only the most elementary 
statistical matters. But we have seen the need of some 
technical knowledge of statistics, which we shall now 
proceed to develop. The treatment will include the 
meanings of the various statistical terms, methods of 
calculating them, cautions as to their use, and devices for 
showing them graphically. 

I. SCALES 

Review of Scales. The first essential in all statistical 
work is to determine the units and scales to be used. It is 
impossible to collect data profitably until this has been 
done. For this reason, these terms were discussed at 
length in connection with the collection of data. If the 
reader is not familiar with the treatment given there,i he 
should read it before proceeding with this section. 

For our purposes here it is necessary to understand : 

1. That whenever possible all the measures in a group should be 
expressed in terms of a definite common measure called a "unit." 

2. That all measures in any group, when arranged in order of size, 
make a scale going from high to low. 

1 See pp. 43-57 
100 



Scales and Distribution Tables 



101 



$170 
160 
150 
140 
130 
120 
110 



Qi47 



20 



10 



University High 169 



Mishawaka 112 



Elgin »00 Maple Lake, Minn. 100 




Norfolk, Neb. 42 

Bonner Springs 38 
Junction City, Kaa 33 ^^^^^^^^^^3^ 



Washington, Mo. 41 

Russell, Kan. 34 



Fig. 6. — Device for Representing a Discrete Scale Graphically. 

This graph shows the cost per 1000 student hours in mathematics in certain high 
schools. (From J. F. Bobbitt, School Review, 23 : 509.) 



102 



School Statistics and Publicity 



3. That we must know what a given measure on a scale means, 
i.e., whether 6 extends from 5.5 to 6.5 or from 6.0 to 6.99 or from 5.95 
to 6.05, etc. 

4. That we must know whether the scale is discrete (all measures 
separate or with gaps between) or continuous (measures running into 
one another and spread out all over the scale). 

It is worth noting that many of the scales the 
superintendent uses, especially those he makes up for him- 
self, are discrete in appear- 
ance but are really used as 
though they were continu- 
ous. That is, each separate 
item is regarded as extend- 
ing half the distance to 
the nearest items above and 
below. 

When the measures are 
discrete and there is a rela- 
tively small number of cases, 
say twenty to thirty, they 
may be shown in a Bobbitt 
table. 1 In this kind of table 
the name of each measure 
is written in the left-hand 
column and the size of the 
measure in the right-hand 
column. The measures begin 
with the highest and run 
to the lowest, thus giving 
an idea of a scale like 
that of a thermometer, 
which goes up from the 
bottom. 
18, 96 




-REVERE- 23.50 
QUINCY- 22.00 
CHEL3EA-21.40 
-ARLlNGTON-20.80 
ICAMBRIDGE- 20.40 

iMELROSE - 20. 40 
EVERETT- 19.70 

IMALDEN -19.20 
WINTHROP- 19.20 
lOMERVILLE- 18.80 
ELMONT- 18.30 
WATERTOWN- 18.20 
MEDFORD- 18.00 
WINCHESTER- 18.00 
NEWT ON -\7. 40 
DEDHAM-17.40 

^BOSTON- 16-4.0. 
^WALTHAM- 15.90 
BROOKLINE- 12.00 
MILTON- 11. 50 



Fig. 7. — Thermometer Chart for 
Presenting a Discrete Scale. 

It shows the total tax on $1000, for the 
year 1912, in all the cities and towns of 
the metropolitan district, Boston, Massa- 
chusetts. Note omission of zero line. 

(From 1912, Newtoji, Massachusetts, 
School Report, page 113.) 

1 See pp. 



Scales and Distribution Tables 



103 



Graphic Presentation of Discrete Scales. A discrete 
scale is easily presented graphically by various methods. 
Four of these methods are given here, three of them using 
the data from the Bobbitt table on page 17. 

1. By vertical scale on left with names of items to the right. 
Figure 6 shows this. 

Cost per 1000 5. Hours $0 30 6o so iso iso leo 

Name of school. 
Uo'verai+y High 
M'Shawaka, Ind. 
Elgin, III. 
Maple Lake,Minn 
Grani+eCity, III. 
Ea3+ Chicago, Ind. 
OeKolb.IlL 
San Antonio, Tex. 
Harvey, 111. 
Waukegqn.TII. 
South Bencl,Ind. 
East Aurora, III. 
Rockford, III. 
Booneville.Mo 
Brazi(,Ind. 
Leavenworth, Kans 
Greens burg. Ind 
Morgan Park, II I. 
Noblesville, Ind 
Norfolk Neb.- 
Washington, Mo. 
Bonner Springs.Kans 
Russell,Kans. 
Junction Cify, Kans 
M+Corroll,III 

Fig. 8. — Graphic«Representation of a Bobbitt Table, Histogram Form. 
The data are from Table 1, page 18. 

2. By thermometer device. 

Superintendent Spaulding in his 1912 report for the schools of 
Newton, Massachusetts, showed a similar table as a scale on a 
thermometer. As this is now out of print, the device is here re- 
produced. It is a very excellent one, save that the zero line is not 
properly shown. See Figure 7. 







! 








$169 












1 


11^ 








1 






100 












100 










88 






1 






82 






1 




74 








1 


b-y 










69 








63 






J 




<62 






1 


61 








59 




J 




58 




1 


56 






56 






54 




J 


53 






52 




f 


42 






41 




1 


s 38 




1 


34 






. 33 






30 

















104 



School Statistics and Publicity 



3. By bars to represent magnitvde. 

The procedure for this is : First, the names of the items (schools 
in this instance) are placed on the left, the highest at the top. In 
the next column is placed the corresponding magnitude (cost per 
1000 student hours in this case). At the top and running out to the 



Cost per 1000 S hrs. ^0 



Name of scbool. 
UniYersi+y HigW ^169 

AAishawaKQ, Ind- I '2 

l\Q,\r\.W. 100 

Map\e Loke, H\ nn. lOO 

Grani+eCi-Vy,IU. 88 

East ChicQCjclnd. 82 

DeKQ|b,Ill. 74 

5a n An^on\o,Tex. 69 

Harvey. l\l. 69 

Waukec^anAVV. 63 

South Bend, \nd. 62 

South Avjrora,W- 61 

RocKtord.lll- 59 

Boonev'\lle,lAo. 58 

Braz.il, \nd. 56 

Leavenworth, Kanv 56 

6reer>\3or«j,\nd. 5^ 

Morgan ParKUl. 53 

Nob\es\'(\\e, \Y\d. 52 

Norto\W,Neto. 42 

Woshincjton.Mo. 41 
Bonner Springs, Kan S. dQ 

Rus3e\\,V^ans. 34 

Junction City, Kans. 33 

Mt.Carro\|,\\\. 30 



30 



60 



90 



120 



150 



180 



^-••^-^—a * ' 



Fig. 9. — Graphic Representation of a Bobbitt Table, Smoothed Curve Form. 

The data are from Table 1, page 18. This is simply the smoothed form of Fig. 8. 



right, the scale is placed (at $10 intervals here). Each distance of 
five small squares on the scale represents $10 and one small square 
will thus mean $2. Therefore, to get the bar for the first school, go 
out on the scale to 170, drop back half a square to get 169, and then 
construct the bar from the base or zero line to this point. In the 



Scales and Distribution Tables 105 

same way the bar for each other school may be constructed. The 
significance of the bar in such a graph lies in its length only, not in 
its width. Hence, all bars in the same diagram must be uniform in 
width. The bars are here drawn adjacent to each other, but there 
can be spaces between if desired. See Figure 8. 

4. By a '^ curve "drawn with the magnitudes on the vertical 
axis and the names of the cases running in order from high 
to low {or vice versa) on the horizontal axis. 

See Figure 9. 

This diagram or graph has been constructed in exactly the same 
way as the preceding one, except that dots have been placed at the 
middle of the spaces where the ends of the bars came in the preced- 
ing diagram, and then joined with a line. The dots were put in 
faintly and have been covered up by the line. Obviously, then, the 
"curve" is nothing more than the smoothing down of the corners 
of the bars. 

EXERCISES 

1. The salaries of school superintendents in Missouri cities be- 
tween 2500 and 5000 population in 1914-15, for all cities on which 
data could be obtained, were as follows : 

Booneville, $1,650 — Butler, $1,320 — Cameron, $1,400 — Carter- 
ville, $1,200 — Caruthersville, $1,200 — Charleston, $1,500 — Clin- 
ton, $1,920 — De Soto, $1,400 — Excelsior Springs, $1,500 — Far- 
mington, $1,400 — Fayette, $1,500 — Festus, $1,140 — Frederick- 
town, $1,450 — Kennett, $1,500 — Kirkwood, $2,400 — Liberty, 
$1,800 — Louisiana, $1,350 — Macon, $1,700 — Marceline, $1,100 

— Marshall, $2,100 — Maryville, $1,500 — Monette, $1,500 — Rich 
Hill, $1,200 — Richmond, $1,500 — Sikeston, $1,500 — Slater, $1,200 

— Warrensburg, $1,600 — Washington, $1,170 — West Plains, $1,500. 
Make up a Bobbitt table to show the status of these salaries, and graph 
this table in as many ways as you can. 

2. Make up a similar table on superintendents' salaries in a group 
of cities in some other state that may be legitimately compared, 
choosing cities of some other size if preferred. 

Statistics on population may be gotten from census reports or a 



106 



School Statistics and Publicity 



good almanac like the World Almanac. Figures for salaries may be 
gotten from directories issued by book companies, from reports of 
the state superintendent of education, or from " A Comparative Study 
of the Salaries of Teachers and School Officers" (Bulletin U. S. Bureau 
of Education, 1915, No. 31). Volume II of the Annual Report of the 
U. S. Commissioner of Education will furnish figures for the total ex- 
pense of the superintendent's office but sometimes this contains other 
items in addition to his salary. In the smaller cities it very closely 
approximates the superintendent's salary. 



n. DISTRIBUTION TABLES 

In a continuous scale or distribution, it is customary 
to group measures of magnitudes that are almost the I 
same in value, and call them by a group name. Thus, 
in the Ayres spelling scale, all words spelled by from 70 
to 76 per cent inclusive of children in a given grade 
are lumped into one group and called 73 in difficulty for 
that grade. 

It is also customary in such a distribution to make up 
a table called a " distribution table " or '' table of fre- 
quency." The table is made up with magnitudes in 
order of size in the left-hand column and the corresponding 
numbers of cases or frequencies in a parallel column on 
the right, the smaller measures preferably being at the 
bottom. The grouping is for the purpose of condensation 
and clearness, but the cases can always be kept individually 
if necessary. Professor Dearborn in a study of grades 
has kept all the cases of each magnitude separate. For 
example, he presents the distribution of marks given in 
English to 69 eighth grade children as shown in Table 
13.1 

1 Dearborn, W. F. : ** School and University Grades," Bulletin, 
University of Wisconsin, No. 368, H. S. Series, No. 9, 1910. From 
Figure 9, p. 25 



Scales and Distribution Tables 



107 



Table 13. Distribution Table of Marks of Eighth Grade 
Children (from Dearborn) 



Mark 


No. Making 


Mark 


No. Making 


100 





80 


3 


99 





79 


3 


98 





78 


3 


97 


2 


77 


2 


96 





76 


1 


95 


2 


75 


2 


94 


1 


74 





93 


3 


73 


3 


92 


4 


72 


2 


91 


6 


71 


2 


90 


2 


70 





89 


3 


69 





88 


3 


68 


2 


87 


3 


67 





86 


1 


66 





85 


2 


65 


1 


84 


3 


64 


1 


83 


2 


63 





82 


3 


62 


1 


81 


3 


61 









60 






The magnitudes in this table run from 60 to 97. This 
table is exactly like a Bobbitt table, except that in this 
instance there are several cases of the same magnitude. 
But even here these cases are supposed or assumed to run 
from the lower limit of the magnitude to the lower limit 
of the next magnitude. That is, the six 91's are not all 
exactly 91 but are spread from barely 91 up to not 
quite 92, or from just 90.5 to almost 91.5, depending 
upon which system of definition of the measure 91 is 
used. 



108 School Statistics and Publicity 

But we know that such grouping as Professor Dearborn 
has used in this instance is rather finer than we are 
accustomed to use in examining teachers' marks. 

Let us now make coarser groupings. The first 
possibiUty that occurs to the reader is probably that of 
making the groupings cover supposedly equal parts of 
the scale, say five units. Thus, we may make the groups 
cover 60-64, 65-69, etc., setting the limits very definitely 
and getting these groups : 



Marks 


No. Making 


95-100 


4 


90-94 


16 


.85-89 


12 


80-84 


14 


75-79 


11 


70-74 


7 


65-69 


3 


60-64 


2 




69 



Or we may make them cover 60-69, 70-79, etc., and 

have: 

Marks No. Making 

90-100 20 

80-89 26 

70-79 18 

60-69 _5 

69 

Or we may group as many school systems do, and have : 



Magnitude 


No. Cases 


95-100 


4 


90-94 


16 


80-89 


26 


70-79 


18 


Below 70 


5 




69 



Scales and Distribution Tables 109 

For grouping, this from Professor Thorndike should be 
kept in mind : 

In general, in mental and social measurements, in the calculation 
of averages, average deviations, and mean square deviations, when 
the face value of the series gives a grouping of 40 to 60 steps, it is al- 
lowable to group by double steps, and when the face value of a series 
gives a grouping of 60 to 80 steps, to group by triple steps. But it 
should be observed that coarse grouping saves little time except in 
the calculation of the average, average deviation, and mean square 
deviation. In the case of the calculation of the median, 25 percentile, 
75 percentile, and median deviation, it is the author's opinion that 
the gain in precision from the finer scale is greater than the loss in 
time, if one economizes time in recording measures in the finer group- 
ing.i 

The superintendent, however, may not understand the 
finer points of this without considerable statistical theory 
and experience. The best simple rule for him to follow 
is not to divide into small groups where the cases seem to 
bunch more closely than usual, and not to include in the 
same group cases that are manifestly far apart. In the 
example of the marks given above, it will be noted that 
the cases bunch together very closely when grouped in 
the third form. Nor are the cases to be found in one 
group too far apart from one another to be in that group. 
For instance, 81 and 89 are to be found in the same group. 
But experience with marks shows us that when one teacher 
marks one boy 81 and another teacher marks another 
boy 89 in the same subject, there may be little appreciable 
difference in the achievements of the boys. 

In most cases common sense and experience must be 
utilized in considering grouping. Salaries of superintend- 
ents may be safely grouped by hundreds (1200-1299, 
1300-1399, etc.) because their salary increases usually 

1 Mental and Social Measurements, p. 50 



110 School Statistics and Publicity 

come by hundreds. But the salaries of grade teachers 
are more profitably grouped by fifties or twenty-fives, 
thus : 400-449, 450-499, or 400-424, 425^49, 450-474, 
474-499, etc., because their usual salary increases are 
covered by the smaller steps. A grouping for training 
in weeks, of teachers that covered summer school work, 
would be 1-6, 7-12, 13-18, etc., or 1-3, 4-6, 7-9, 10-12 
and not 1-4, 5-8, 9-12, because summer schools usually 
run either 6, 9, or 12 weeks. 

EXERCISES 

1. The salaries of superintendents in cities of 2500-5000 for 1914- 
15, for all obtainable, were in the states given, as follows : 

Alabama. $1,500 — 1,800 — 1,250 — 1,680 — 1,900 — 1,600 — 

1,800 — 1,500. 
Arkansas. $2,000 — 1,620 — 1,000 — 1,500 — 1,600 — 1,600 — 

1,500 — 1,100 — 1,500 — 1,200 — 1,500 — 1,500 — 1,600 — 

1,350. 
Florida. $1,500 — 1,200 — 1,500 — 1,650. 
Georgia. $1,800 — 1,800 — 1,800 — 2,000 — 1,500 — 1,200 — 

1,500 — 1,500 — 1,200 — 1,600 — 1,800 — 2,000 — 2,000 — 1,650. 
Kentucky. $1,000 — 1,500 — 1,800 — 1,350 — 1,800 — 1,600 — 

1,800 — 1,400 — 1,200 — 1,400 — 1,650 — 1,200 — 1,400 — 

1,500. 
Louisiana. $1,500 — 1,800 — 1,500 — 1,800 — 1,500. 
Maryland. $1,400 — 1,450. 

Mississippi. $1,800 — 2,200 — 1,125 — 1,700 — 1,650 — 1,800. 
Missouri. $1,650 — 1,320 — 1,400 — 1,200 — 1,200 — 1,500 — 

1,920 — 1,400 — 1,500 — 1,400 — 1,500 — 1,140 — 1,450 — 

1,500 — 2,400 — 1,800 — 1,350 — 1,700 — 1,100 — 2,100 — 

1,500 — 1,200 — 1,500 — 1,500 — 1,200 — 1,600 — 1,170 — 

1,500. 
North Carolina. $1,500 — 1,500 — 1,200 — 1,500 — 1,200. 
Oklahoma. $1,500 — 1,500 — 1,400 — 1,800 — 1,800 — 1,500 — 

900 — 2,000 — 1,300 — 1,800 — 1,500 — 1,800 — 1,200 — 1,800 

— 1,300 — 1,500 — 1,800 — 1,800. 



Scales and Distribution Tables 111 

South Carolina. $1,200 — 1,500 — 1,215 — 1,500 — 1,350 — 
1,800 — 1,200 — 1,250 — 1,500 — 2,000. 

Tennessee. $2,000 — 1,000 — 1,200 — 1,080 — 1,200 — 1,600 — 
1,600 — 1,500 — 1,000 — 1,800. 

Texas. $1,960 — 2,000 — 1,500 — 1,800 — 2,100 — 2,200 — 
2,000 — 1,800 — 1,500 — 1,800 — 2,000 — 1,200 — 1,500 — 
1^560 — 1,800 — 1,500 — 1,800 — 1,300 — 1,800 — 2,300 — 
1,500 — 1,675 — 2,200 — 1,200 — 2,000 — 1,400 — 1,800 — 
1,500 — 1,500 — 1,500 — 1,800 — 1,800 — 1,800. 

Virginia. $1,750 — 1,200 — 1,200. 

West Virginia. $1,500 — 1,400 — 1,350 — 1,500 — 1,500 — 1,800 
— 1,500 — 1,380 — 1,600 — 1,550 — 2080. 

Ignore the matter of sampling and arrange these salaries in a distri- 
bution table, being careful to justify the step chosen in your grouping. 

2. Make a similar distribution table of superintendents' salaries 
in cities of this size for any other section of the United States, getting 
your data from the sources found in Exercise 2, page 105, and telling 
just why you use each step in the process of making the table. 

3. Make a similar distribution table for the following figures on 
the number of hours required by individual pupils to complete one 
half-grade in grammar : ^ 

7 — 10 — 11 — 11 — 11—12 — 12 — 13 — 13 — 15 — 16 — 16 
_ 16 — 17 — 18 — 18 — 19 — 19 — 20 — 20 — 21 — 21 — 22 — 
22 — 22 — 23 — 23 — 25 — 27 — 29 — 33 — 33 — 33 — 34 — 34 
_ 36 _ 37 _ 38 — 39 — 40 — 43 — 44 — 44 — 48 — 49 — 49. 

III. SURFACE OF FREQUENCY 

Graphing Distribution Tables. For the presentation of 
grouped distributions by graphs, three simple devices 
are available, the histogram or rectangular graph, the 
smoothed graph, and the check form of the histogram. 

The procedure for this histogram is as follows : 

1. Lay off on cross-section paper a horizontal scale, on which the 
magnitude scale runs by groups from the lowest magnitude at the left 
to the highest magnitude at the right. 

1 From Monograph C, Individual Instruction, San Francisco State 
Normal School, p. 28 



112 



School Statistics and Publicity 



2. From the same zero point erect a perpendicular scale which 
is to represent the number of cases. 













r— 




















< 














25 


























20 
























\ 
1 


























• 










y 










15 


























10 










































5 : 






















I 


1 
1 

1 

1 










\ 










' 








1 

1 
1 










1 
1 

1 





"No. Below 
Cas65 70 



70 t079 80 to 89 SO to 84 QSfoWO 



Fig. 10. — Histogram Showing Data of Table 13, but Using Grouping as 

Given on Page 108. 



3. Then find on the horizontal scale the point marking the mag- 
nitude of any given case, and count up to find the proper point to 
denote the number of cases in that group. 



Scales and Distribution Tables 



113 



Do the same for each group. In so doing, one will get a number of 
points at different heights strung out above the horizontal scale. 

4. Then proceed to draw a line through these points coming down 
to the base line on the right, and either coming down to the base line 

























25 










j 






















/ 












70 










/ 


































(5 






/ 








> 


^ 












/ 










\ 






/O 




/ 


' 










\ 








/ 














\ 




5 




^ 














\ 






/ 
















\ 


\ 




/ 


















\ 



Cases 70 

Fig. 11. — Smoothed Form of Graph Shown in Figure 10. 



70 to 79 ao f o 69 00 to 94 9 5 f o 100 



on the left or going to the vertical scale on that side. There will thus 
be inclosed an area which is called the "surface of frequency." 

This surface may be made by making each point located after the 



114 School Statistics and Publicity 

manner described above, the upper left-hand corner of a rectangle 
which is as wide as the length of the space occupied by that group on 
the horizontal scale. Thus the plotting of the Dearborn data as a 
histogram is shown in Figure 10. 

It is not customary to draw those parts of each rectangle 
shared in common with other rectangles. Common por- 
tions in the diagram are shown by dotted lines. 

In the smoothed graph (Figure 11), the points located 
to determine the rectangles may represent the middles 
of the tops of such rectangles instead of the upper left- 
hand corners. Then these points may be joined by straight 
lines, giving a surface with apexes. This is somewhat 
" smoothed," it will be noticed. 

The check form of the histogram simply uses dots on 
cross-section paper, one for each item, thus keeping the 
columns the same width. There is no line drawn above, 
the columns showing roughly the shape of the surface. 
It is a very valuable form for tabulating data and at 
the same time showing the shape of the surface of fre- 
quency. That is, it may be made up before the dis- 
tribution table. Thus, if cross-section paper had been 
used for the. Dearborn data at the outset, a surface of 
frequency like that in Figure 12 could have been obtained, 
and from this surface, the distribution table given on page 
107 could have been easily made up. 

In graphing a distribution table, the scale in which the 
items come regularly is always put on the base line. 
The scale in which the items come more or less irregularly 
is put on the vertical line. In other words, this means 
that magnitude is measured on the horizontal line of the 
graph and the number of cases is shown on the vertical 
line. There is no bullet-proof reason for this ; it is simply 
the conventional way of doing the thing, just as the order 



Scales and Distribution Tables 



115 



























... 




n 






















t3 
O 






























0) 






























QO 


























— 


— 


K« 






























(0 


























— 


— 


lO 




























— 


^ 
























— 


~ 


— 


«^ 






















— 


— 


— 


— 


CM 


















— 


— 


— 


— 


— 




~" 


























— 


— 


























— 


"~" 




<n 
























— 


— 


— 


CO 
























— 


— 


— 


t- 




























— 


<0 


























— 


— 


«o 
























— 


— 


— 


"t 


























— 




ro 
























— 


— 


— 


rvJ 
























— 


— 


— 


""* 
























— 


— 


— 


O 
flO 
























— 


— 


— 


O) 
























«. 


•— 


— 


CO 


























— 


— . 


f- 




























— 


<0 


























— 


~ 


ir» 






























t 
























— 


— 


— 


n 


























— 


— 


CJ 


























— 


— 


"* 






























^ 






























o 


























— 


— 


00 






























h- 






























<d 




























— 


tn 




























— 


^ 






























n 




























— 


CM 






























"^ 









































a 



o 




o . 




OS (H 




V o 








^S 




03 •-< 




-t-j 




O ^ 




3co 


CO 
i-H 


^03 t}2 

Eh ^ 


3 


_. »-* 


Sj 


HO 


H 


•43 fl 


a 


P o 


o 


-Q w 


y-i 


•S ^ 


a> 


4^ W 


^ 


03 (S3 


c4 


«^ 


1^ 

+3 


bD 


rt 


M.S 


■73 




1 


(D r£l 




CO 




(H >i 




o-o 





to 



f*< 



116 School Statistics and Publicity 

from left to right across the page and from top to bottom 
in reading and writing is the proper and customary one 
for European people and their descendants. One should 
draw pictures or graphs of data to be read with as much 
care as he would exercise in preparing the manuscript of 
an article to be printed. ^ 

Characteristics of Surface of Frequency. The dis- 
tribution table is a great economy over a miscellaneous 
mass of unassorted data. But it is too cumbersome to 
be kept in mind in all its details. We need to apply here 
our idea of the fundamental image or bird's-eye view of 
the whole. This can best be done through the use of 
two special qualifications, characteristics, or earmarks of 
the distribution. The first of these qualities is that 
measure which indicates the typical, average, or central 
size of the group. The second is that number which 
indicates how far the other members of the group on the 
average vary, spread, or deviate from the first-named 
quality. These two characteristics of the distribution 
table serve to make it full of meaning to any person who 
understands statistics; and with a little care the lay 
reader may readily acquire the ideas back of these 
devices. 

Both of these characteristics may be expressed as magni- 
tudes or as so many multiples of whatever unit may be 
used in the distribution table. They may also be shown 
graphically on the surface of frequency. The particular 
kind of central tendency to be used and the measure from 
it depend altogether upon the shape of this surface of 
frequency. The question at once comes up, then: 
How many variations in the surface of frequency are of 

1 Paraphrased from W. C. Brinton : Graphic Methods of Present- 
ing Facts 



Scales and Distribution Tables 117 

significance to the superintendent, and how may he know 
them? 

Normal Surface of Frequency. If the cases in any 
distribution are taken by chance or by a combination of 
causes that amount to chance, the shape of the surface 
of frequency invariably becomes bell-shaped with symmet- 

Percenf 
inaKirig 
given score 




Lowesf Highest 

score score 

Fig. 13. — Example of Normal Surface of Frequency. This graph shows 
the per cent of pupils attaining given scores in Stone Reasoning Tests, all 
pupils being tested. (Adapted from Butte Survey, page 95.) 

rical sides. Figures 13, 14, and 15 are good examples 
because all the children were taken, and the variations 
then arise only from chance. 

Notice that in one of these graphs or diagrams the cases 
composing it are bunched much more closely around the 
highest point or apex of the distribution than in the 
second diagram. But it might sometimes happen that 
two different sets of children, when tested on the same 
thing, might give distribution tables which, when graphed, 
would make graphs or surfaces of frequency varying as 
widely as these two. For example two sixth grades 
might make the same average or central tendency on the 
Courtis Tests. But in one case the achievements of all 



118 



School Statistics and Publicity 



the children might be close to the average, while in the 
second case some of the children might make very high 
records and others very low. This bell-shaped surface is 
called the " normal " or " probability " surface because 
it is the one found in natural and mental phenomena of 
all kinds, when the distributions are made up from un- 



5043 



4462 



3536 



2878 



1373 



452 



2 59 



3536 



1884 



1225 



527 



399 



10 20 30 40 50 60 70 80 90 100 110 120 
to to to to to to to to to to to to to 
9 19 29 39 49 59 69 79 89 99 109 119 129 

Fig. 14. — Example of Normal Surface of Frequency. This figure shows 
the number of pupils writing at each speed from to 9 letters per minute 
to 120 to 129 letters per minute. Data for 25,387 pupils in four upper 
grades of Cleveland. (From Measuring the Work of Public Schools, Cleve- 
land Survey, page 67, by permission.) 



selected cases or from those picked at random. In other 
words, it is the normal or probable thing to expect from 
such data. 

Skew Surface of Frequency. But there are some 
distributions in which the cases bunch much more on one 
side of the apex than on the other, for the reason that a 
certain cause or combination of causes operates on some 
of the cases in the distribution but not on all. The 



Scales and Distribution Tables 119 

distribution or surface is said to be " skewed " toward 
the thin, drawn-out side and is called a '' skew '' dis- 
tribution or '' skew " surface. ^ A good example of such 
a distribution is one made up of the number of children 
of different ages in school. As the children grow older, 
some die, and consequently the group of a given age is 



5000- - 




Fig. 15. — Smoothed Form of Figure 14. 

slightly smaller in size than any group of lower age. But 
at the age of fourteen the compulsory education laws 
usually cease to operate, and many children immediately 
drop out of school. Others want to go to work. Such 
forces tend to decrease the group rapidly. The follow- 
ing table of frequency for 1908 for Nashville, Tennessee, 
as reported by Stray er in his " Age and Grade Census of 

1 From the verb "skew," meaning to put askew or twist to one side. 
While there is perfect agreement among educational writers as to what 
constitutes a skew distribution, there is no agreement as to which is 
the skew end. By this term some writers mean the blunt end and 
some the sharp end. The author uses it to describe the sharp end, 
following Professor Thorndike. 



120 



School Statistics and Publicity 



Schools and Colleges," ^ well illustrates the point under 
discussion : 

Table 14. Distribution by Ages of Pupils in the Nashville 

Schools, 1908 



\ 


Age 


Number in group 




72 


1493 




8 


1733 




9 


1584 




10 


1712 




11 


1595 




12 


1626 




13 


1490 




14 


1198 




15 


885 




16 


528 




17 and over 


390 



Notice that there is practically no increase or decrease 
of importance in these groups until the age of fourteen 
is reached. From this point on, the decrease is very 
rapid. If the ages beyond seventeen had been given sep- 
arately, the extension would narrow down to a very 
slender one. Figure 16 shows the facts in the histogram 
or unsmoothed form. 

Smoothed out, as in Figure 17, the graph shows the 
" skew " even better. 

A good example of a skew surface is shown in the results 
of the spelling test in Figure 18. 

1 Bulletin U. S. Bureau of Education, 1911, No. 5, p. 34 
' Children are not permitted to enter the Nashville schools until 
they are seven years of age. 



Scales and Distribution Tables 



121 



< 1 



Mo, of 

pupils 

1800 
1600 

1400 
/200 
1000 
600 
600 
400 

^00 



A36S 6 7 8 10 II 12 13 14 15 \Q 17 and over 

Fig. 16. — Skewed Histogram Representing Distribution of Pupils by 
Ages in Nashville Public Schools, 1908. (From data in Table 14.) 

No. of 
pupils 
1800 

1600 



1400 
1200 

(000 
800 
600 
400 
200 



Ages 6 7 8 S 10 II 12 13 14 15 16 17 and over 
Fig. 17. — Smoothed Form of Graph Given in Figure 16. 



-V 

1 — I ^ 

i I I I I I I I I I 



122 



fo Of children 
40 A 



School Statistics and Publicity 

M91 



30- 



20- 



10- 



ENTIRE CITY 
3988 cinildren 



I 



10 20 30 40 50 60 -70 80 90 100 

Fig. 18. — Skewed Histogram Showing the Percentage of Children 
Attaining Each of the Possible Scores on the Spelling Test in Salt Lake City 
as a Whole. (Adapted from Salt Lake City Survey, page 135.) 

EXERCISES 

1. Draw a surface of frequency for each of the distribution tables 
used or gotten up by you in the previous exercises. 

2. Draw a surface of frequency for each of the distributions given 
in the table on page 123. 

REFERENCES FOR SUPPLEMENTARY READING 

King, W. I. Elements of Statistical Method, Chapters V, XI. 
Rugg, H. 0. Statistical Methods Applied to Education, Chapters 

IV, VII, VIII. 
Thorndike, E. L. Mental and Social Measurements, Chapter II and 

pages 28-36. 



Scales and Distribution Tables 



123 



'Frequency of the Different Percentages of Boys and 

Girls Retarded Two Years in Certain Cities of 25,000 

Population and Over, 1908 ^ 



Per cent of total 
no. of hoys 


No. cities 


Per cent of total 
no. of girls 


No. cities 


2 


1 


2 


3 


3 


5 


3 


6 


4 


3 


4 


9 


5 


7 


5 


9 


6 


7 


6 


9 


7 


4 


7 


12 


8 


9 


8 


18 


9 


16 


9 


15 


10 


12 64 


10 


13 


! 11 


12 


11 


11 


12 


19 56 


12 


7 


13 


11 


13 


3 


14 


9 


14 


2 


15 


2 


15 


5 


16 


2 


16 


3 


17 


1 


17 


3 


18 


7 


18 


2 


19 


1 


19 


2 


20 


2 


21 


1 


21 


1 






22 


1 







1 From Strayer, G. D. : "Age and Grade Census of Schools and 
Colleges," Bulletin U. S. Bureau of Education, 1911, No. 5, pp. 86-87 



CHAPTER V 

MEASURES OF TYPE 

There are three measures of the " type " or central 
tendency of importance in school work — the mode, the 
median, and the average — which will now be discussed 
in order. 

I. THE MODE 

Definition. The mode is that number which represents 
the size of the most numerous item or items in a group. 
That is, it is the vogue or fashion in the cases, because 
there are more of this size than of any other. The mode 
is precisely what the ordinary man usually has in mind 
when he speaks of the '' average.'' He is referring to that 
measure which includes the greatest number of cases. If 
he says that teachers instruct forty children on the 
average, he means that more teachers instruct just about 
forty children than teach thirty, fifty, or any number 
far removed from forty. 

Graphic Representation. Graphically, the mode is 
the magnitude represented by the point on the scale 
above which the surface of frequency is highest. It may 
be marked by a perpendicular erected from this point 
to the apex of the surface. But note that the mode is a 
measure, not a number of cases. 

Calculation. All that is necessary here is to pick out 
the group containing the largest number of cases and 

124 



Measures of Type 



125 



see what its magnitude is. If several adjoining groups 
have about the same number of cases in them, they 
should be run together into larger groups. Usually this 
procedure will give a more pronounced mode, which is 
its purpose. In case two widely separated groups are 
larger than the others, the distribution is said to be " bi- 

Number receiving 

1251 



iOO 



15 



50 



25 



n 



^ 






^^ 



,0 (O 



o 



" 0) — > >- V. 






o O 



^ <x> 



o 

0; 



Salary 




C\J 



C 

Q> 



CO 






O 



^ooo o ooO(io oo ooSoo 



Fig. 19. — Example of Multi-modal Surface of Frequency. 

This shows the distribution of salaries paid elementary school teachers, Salt Lake 
City, 1914-15. Note that the printing on this graph cannot be read easily from one 
position. (Adapted from Salt Lake City Survey, page 51.) 



modal, ^' that is, it has two modes, and probably is 
composed of two rather different classes which have been 
lumped together but which possibly should not be so 
considered. A frequency table showing the number of 
teachers getting different salaries will often show more 
than one mode. For example, the surface of frequency 



126 School Statistics and Publicity 

for the salaries of elementary teachers in the Salt Lake 
City Survey has three well-defined modes at approximately 
$650, $850, and $1020. This means that, as regards 
salaries, there are really three distinct classes of teachers 
within the whole group. The graph is here shown in 
Figure 19. 

Advantages of the Mode for School Statistics. 1. The 
mode is useful where it is desirable to eliminate extreme 
variations. 

For example, the amount of work a given group of children can do 
in a school year is determined by the modal attendance of the group, 
not by that of the few who are absent almost continuously, or that 
of the small number who never miss a day. 

2. In finding the mode, it is unnecessary to know 
anything about extreme cases except that they are few 
in number. 

In comparing his school with other schools, the superintendent 
need not worry about the one or two schools that are higher than any 
of the others, if his own school falls in the mode or close to it. The 
extreme cases may not be measured accurately, and they may or may 
not really come legitimately into the distribution. But whether 
they do or not, they cannot affect the mode. 

3. It is very easy to determine with considerable 
accuracy from well-selected data. 

4. It is the best measure of type to the ordinary mind. 

As before indicated, this is what the ordinary man often means by 
"average." 

5. It is unambiguous. 

No one ever thinks from it that all the measures in the group are 
practically on it. 

6. The mode is often the most typical measure of a 
skew distribution. 



Measures of Type 127 

Probably the most significant thing about a frequency table of 
teachers' salaries is that largest group which get the same salary or 
a salary within certain limits. The extreme salaries, the median 
salary, or the average salary might be of no especial significance. But 
the modal salary would point out the significant group at once. The 
three modal salaries for the Salt Lake City elementary teachers, as 
shown in the graph on page 125, indicate the significant salaries at a 
glance. 

Disadvantages of the Mode for School Statistics. 1. In 

many groups, no single, well-defined type actually exists. 

There is no such thing as a modal age for children or a modal num- 
ber of children in a grade in school. All the age groups up to 14 are 
about the same in size, and all the grades up to about the sixth or 
seventh keep about the same size. Of course children drop out but 
usually enough are held over to make the grades approximately the 
same size. 

When all cases are kept separate as in Bobbitt tables, there is, of 
course, no mode unless several cases happen to be of exactly the same 
size or are considered to be of the same size. 

2. The mode is of no value if weight is to be given to 
extreme cases. 

It would take no special account of the high per capita cost of a 
city at the upper end of a per capita group, so far as the size of the 
item was concerned, although such city might admittedly have the 
best schools in the group. Similarly it would take no special note of 
the lowest city in the group although it might admittedly have the 
worst schools in the group. 

3. The mode cannot be determined by any simple 
arithmetical process and is sometimes difficult to get by 
any method. 

4. The product of the mode by the number of items 
does not give the correct total. 

For example, take Table 15. 



128 School Statistics and Publicity 

Table 15. Distribution Table Showing Penmanship 
Records of Second Grade at Butte ^ 

Score ■ Number pupils making 

4 5 

5 22 

6 21 

7 29 

8 28 

9 42- 

10 7 

11 29 

12 5 

13 7 
16 _1 

196 

The mode in this example is 9. 196 X 9 = 1764. The sum of 
the products of the cases in each group by its score, however, is only 
1617. Such a total sometimes proves very useful for checking other 
steps. 

5. The mode may be determined by a very few items 
in case none of the groups contains more than a few 
items. 

This, of course, may be offset by wider grouping. 

EXERCISES 

1. What is the mode in each of the distribution tables used in pre- 
vious exercises ? 

2. Draw the line to represent the position of the mode on each of 
the surfaces of frequency used in previous exercises. 

II. THE MEDIAN 

Definition. The median is the magnitude represented 
by the mid-point on a scale or distribution. Obviously, 
half the cases fall below this mid-point and half above it. 

1 Butte Survey, p. 80 



Measures of Type 129 

Note that the median is a magnitude or size of a case, 
not the number of the case. 

Calculation. Various devices and formulas have been 
given for calculating the median, according as there is an 
odd number of cases, an even number of cases, a gap 
between two groups that are equal in size, etc. But the 
simplest and surest plan hy far is to regard the distribution 
as a scale and always to find the magnitude of the mid-point 
on it. 

If there is an odd number of cases, the median is, of 
course, where the mid-point of the middle case lies. If 
the median falls in a gap, the mid-point in the gap must 
be taken. If the median falls in a group distributed 
over part of the scale, one must run up the part covered 
by this group until he finds the point that will exactly 
place half the cases in the whole distribution below it 
and half above it, splitting a case into halves if it is 
necessary. 

The essential thing is to find the mid-point. The mag- 
nitude denoted by this mid-point will be the exact median. 
The magnitude corresponding to the group containing the 
mid-point will be the approximate median. This mid- 
point method of calculation will now be illustrated with 
various examples, starting with an even number of cases. 

If the distribution is a discrete one and contains an 
even number of cases, the median falls between the two 
middle cases. The place for it to fall is found by dividing 
the number of cases by 2, which gives the number of 
cases to have on one side. Then count in from one end 
till this number of cases has been checked off. 

For instance, suppose we make up a Bobbitt table with an even 
number of cases by taking only the first twelve cases from the table 
on page 18, as in Table 16. 



130 



School Statistics and Publicity 



Table 16. Bobbitt Table Showing Cost of Instruction per 
1000 Student Hours (Mathematics) 



Name of school 



University High 
Mishawaka, Ind. . 

Elgin, 111 

Maple Lake, Minn. 
Granite City, 111. . 
East Chicago, Ind. 

De Kalb, 111. ^ 
San Antonio, Tex. . 
Harvey, 111. . . . 
Waukegan, 111. . . 
South Bend, Ind. . 
East Aurora, 111. 



Cost per 1000 
student hours 



$169 
112 
100 
100 

88 

82 



74 
69 
69 
63 
62 
61 



The mid-point of these twelve cases will obviously have to throw six 
cases on each side of it, that is, it must come between cases six and 
seven. For those who desire a formula, this will be found by dividing 
the number of cases by 2. The mid-point, then, is at the magnitude 
halfway between $82 and $74 or at $78. ($82 - $74 = $8. h of $8 
= $4. $74 + $4 = $78.) The same result, of course, could be 
obtained by merely taking the average of the two middle cases. 
($74 + $82 = $156. $156 ^ 2 = $78.) 

If the two cases between which the median in a Bobbitt table falls 
are the same size, as in the Bobbitt table of size of classes on page 96, 
the median, of course, is represented by the size of either case, — 19 in 
this instance. 

For an example of a continuous series and even number 
of cases, take the achievements of the eighth grade in 
composition at Butte. ^ 

These may be adapted for our purposes as follows, paying attention 
for the present to the two left-hand columns only : 

1 Butte Survey, p. 74 



Measures of Type 131 



Rated at 


Number of ; 


papers 


7 


2 




6 


6 


(8) Adding down 


5 


22 


(30) 


i4 


43 


(73) 


^^3 


39 




^ 


32 


(42) 


1 


9 


(10) Adding up 





1 





2 )154 

77 

There are 154 cases in all. The median, then, must be at the point 
which will throw 77 cases on one side and 77 cases on the other. This 
point will obviously be at the end of the 77th case or the beginning of 
the 78th case, as one prefers to call it. Say that this point will be 
located at the end of the 77th case. Counting up we find 42 cases in 
steps 0, 1, and 2. If the 39 cases in step 3 are added, we find that we 
have more than the required 77 cases. Subtracting 42 from 77 we 
find that we must have 35 cases more. That is, we must add 35 cases 
from group 3 to the other 42 so as to reach the end of the 77th case. 
This means that the median is located |f of the distance up the scale 
represented by the step 3. (ff = .90) 

But does step 3 extend from 3 to 4, or from 2.5 to 3.5? If it ex- 
tends from 3 to 4, the median is obviously 3.90. (3 + .90 = 3.90) 
In the Butte scoring, however, the latter method was used, and the 
actual values on the Hillegas scale are as follows : 

is 3 is 3.69 

1 is 1.83 4 is 4.74 

2 is 2.60 5 is 5.85, etc. 

With this in mind, we must use the true measures on the Hillegas 
scale. Step 3 would extend from the halfway point between 2.60 
and 3.69 (or 3.145) and the mid-point between 3.69 and 4.74 (or 
4.215). The distance between 3.145 and 4.215 is 1.07. f| of 1.07 
is .96. Adding this .96 to 3.145 we get the median or mid-point, 4.105. 
The median may also be figured from the other end. The pro- 
cedure is the same, the figures this time being as follows. Counting 
down we find that from quality 4 up we have 73 cases. We need to 
take 4 cases from the upper part of the group rated 3. That is, we 
must go down /^ of the part covered by that group, (/j = .10) If 



132 School Statistics and Publicity 

the step meant 3 to 4, the median would then be 4 — .10 or 3.9. But 
as we saw before, this group covers 1.07 and extends up to 4.215. 
3% of 1.07 is .11. Then 4.215 - .11 gives 4.105 for the median. 

If the distribution is a discrete one and contains an odd 
number of cases, the magnitude of the middle case is the 
median. This is easily found by counting in, usually 
adding 1 to the number of cases and dividing by 2 to 
get the number of the middle case. 

Thus, in the discrete series represented by the first two columns 
of the table on page 51, the median real wealth behind each $1 for 
schools is $234, because it is the eighteenth case on the scale. There 
are 17 cases below and 17 cases above. Note that 18, the number 
of case wanted, is found by adding 1 to the total number of cases (35) 
and dividing by 2. (1 + 35 -^ 2 = 18.) In the Bobbitt table on page 
18, the median is represented by the thirteenth case. (13 = 1+25 -^2.) 
Note the horizontal lines inclosing the median. 

If, however, the series is a continuous one and has an 
odd number of cases, the median is manifestly located at 
the mid-point of the middle case. In the discrete series, 
we took the whole middle case for the median. In the 
continuous series, the middle case is itself supposed to be 
spread out along the scale, and consequently we have to 
find its mid-point. 

For example, let us take the median for the fifth grade composi- 
tion scores at Butte.^ These were as follows : 



Rated at 
5 


Number making 

1 


4 
3 
2 


18 
49 
86 


(19) Adding down 
(68) 


1 




46 

1 

2)201 


(47) Adding up 



100.5 
Butte Survey, page 74 



Measures of Type 

This time we have an odd number of cases. The median will fall 
on that point where 100| cases come on either side, that is, in the 
middle of the 101st case. In steps and 1, we have 47 cases. We 
need 53 § cases out of the 86 in step 2 to find our halfway place. Chang- 
ing step 2 to the Hillegas value as before, we find that it extends from 
the mid-point between 1.83 and 2.60 to the mid-point between 2.60 

and 3.69, that is, from 2.215 to 3.145, the distance being .93. ^^ 

86 
of .93 is .58. 2.215 plus .58 makes 2.795, the median. 

Coming down, we find that we have 68 cases from 3 up. To get 

100| cases, we must take 32^ cases from the upper end of group 2. 

Figuring as before, ^ of 93 = .35. 3.145 - .35 gives us 2.795 
86 

for the median, the same result as before. 

Some books give rules for finding the median which 
involve finding the middle case. The middle case does 
represent the median in that its magnitude is the median. 
But there is danger in using the formula of adding 1 to 
the total number of cases and dividing by 2 to get the 
middle case. The danger lies in the tendency to add the 
whole of the middle case to the part taken from the 
group. Thus in the example preceding, a beginner is apt 
to take fl of step 2 to add to the lower limit of that 
step, or to take || from the upper limit if he is coming 
down. The former procedure really puts 101 cases below 
and only 100 cases above the median calculated. The 
other procedure puts 101 cases above and only 100 cases be- 
low the median calculated. Obviously, two different re- 
sults would come from these two calculations. That is, 
the median calculated coming down would not agree with 
the median calculated coming up. The mid-point plan, 
however, will give the same result from either end and 
is consequently safer. 

In order to emphasize the fact that the median is best 
located by taking the mid-point on the scale or dis- 



134 School Statistics and Publicity 

tribution, irrespective of whether the number of cases is 
odd or even, let us take several other examples. 

In 1916, the Courtis Tests were given in certain schools in a western 
city, with the following results for the number of problems attempted 
by each eighth grade child in one process : 

No. 



problems attempted 


No. attempting 


24 


5 




23 


2 


(7) Adding down 


22 


2 


(9) 


21 


1 


(10) 


20 


5 


(15) 


19 


5 


(20) 


18 


5 


(25) 


17 


7 


(32) 


16 


13 


(45) 


15 


15 


(60) 


14 


10 


(70) 


13 


15 


(85) 


12 


24 


(109) 


11 


22 




10 


26 


(88) 


9 


21 


(62) 


8 


19 


(41) 


7 


9 


(22) 


6 


8 


(13) 


5 


3 


(5) 


4 


1 


(2) Adding up 


3 


1 
2)219 
109.5 





There are 219 cases. Therefore, the median will fall in the middle of 
the 110th case, since there must be 109 § cases on either side of it. 
Steps 3 to 10 inclusive take in 88 cases. Subtracting 88 from 109.5, 
we find that we must have 21.5 cases out of the next step. There are 
22 cases in the step. Therefore, ^ of the distance of this step, 
which is 1, is .98. By the nature of the Courtis Tests, however, step 



Measures of Type 135 

11 extends from 11 to 12o Therefore, the median is 11 + .98 
= 11.98. Figuring down, we would have ^ of 1 = .02 to be taken 

from the upper limit of step 11, which is 12. 12 — .02 = 11.98 for 
our median, the same result as before. 

Suppose, now, that we manipulate the data in the above problem 
so that step 11 contains only 1 case instead of 22, the remaining 21 
being given to step 10, making its total 47. The total of all the 
cases is unchanged. We find, however, that steps 3 to 10 inclusive 
now contain 109 cases, and the median must fall in the 11th step, 
which has only one case. It will, therefore, fall in the middle of the 
step, since the one case must be considered as extending over the whole 
space in the step. The median, therefore, in this supposed case would 
be 11.5. 

In order to show the working of the same principle in the case of 
the fifth grade composition results at Butte, given above, let us ma- 
nipulate the data until they appear as follows : 



Rated at 


No. Papers 


5 


23 


4 


28 (51) Adding down 


3 


49 (100) 


2 


1 


1 


46 (100) Adding up 





54 




2)201 




100.5 



The total number of cases remains the same, and the median would 
fall as above, namely, at the middle of the 101st case, or at the 100| 
case. Steps and 1 now together contain 100 cases. Therefore, h 
case must be taken from the next step. As there is only one case in 
this step, the median will fall at the mid-point of this step, for the same 
reason as stated in the solution of the problem just above this one. 
The distance covered by step 2 on the Hillegas scale, as shown before, 
is .93. One-half of this is .465. Add .465 to 2.215 (the lower limit 
of the step) and we have for the median, 2.68. 

Suppose, however, that we have an even number of cases, and the 
median falls in a step containing cases, that is, in a gap in the dis- 
tribution. For example, let the data used in the last problem be ma- 
nipulated so as to appear thus : 



136 School Statistics and Publicity 



ed at 
5 


No. papers 
23 


4 
3 
2 


28 (51) Adding down 
49 (100) 



1 



46 (100) Adding up 
54 




2)200 
100 



In this example, the median would come at the end of the 100th case. 
We find that the end of this case coincides with the end of step 1. 
But coming down, the end of the 100th case would be at the begin- 
ning of step 3. Obviously any point in step 2 could be taken and 
there would still be 100 cases on each side. In a problem of this 
kind, the best procedure is to divide the vacant space on the scale 
so that the two nearest actual cases on it will be equidistant. Then 
the median will fall in the middle of step 2 or at 2.68 for the real value, 
and will be the same as if one case had been in the step with the total 
number of cases odd. 

If the data given on page 134 from the tests in arithmetic are ma- 
nipulated so that there are only 218 cases in all and one-half of these 
are above 11 and one-half below, with none in step 11, but actual 
cases in steps 10 to 12, precisely the same procedure would be followed. 
That is, in this case, the median would be 11.5. 

Sometimes the median is not calculated exactly but 
only approximately. Professor Monroe, for example, 
gives some devices for obtaining an approximate median 
and then correcting it to get the true median.^ But in 
general, about the only time that an approximate median 
may be used to advantage is when one takes for it the 
magnitude of the lower limit of the group that contains 
the real median. That is, 11 would be taken for the 
approximate median for the problem on page 134, since 
the median falls within the 11-11.99 group. The straight 

* Monroe, W. S. : Educational Tests and Measurements, pp. 242- 
247 



Measures of Type 



137 



mid-point method of calculating the median will get it 
with less confusion, greater accuracy, and more speed for 
the average person. 

Graphic Representation of the Median. In a Bobbitt 
table having an odd number of items, the median is the 
No. attempting 



ow 








































































1 


















































1 


















































1 

1 




























25 










































































































































































































































































































zo 


























































































































































































































































\D 












































































































































































1 


















































1 


















































1 


















































1 


















































1 


















































1 




























5 






















1 
















































1 


















































) 


















































1 


















































1 
1 
















































1 


























( 

No. Pr 


Dbli 


sm 


s 




1 


& 




G 


9. 

juar 


'0 II. 

Medi 

65 

tile 1 


98 
an 

2G 


15 

15. 

Quar 
^ 


35 

file 


3 




2 











2 


5 



Fig. 20. — Graphic Representation of the Median on a Surface of Frequency. 
The corresponding distribution table is given on page 134. 



138 School Statistics and Publicity 

size of the middle item, and it is usually inclosed in parallel 
horizontal lines to indicate that it is the median. (See 
page 18.) If the table has an even number of items, a 
single horizontal line is drawn across between the two 
middle cases, thus throwing half the cases above and half 
below. (See page 130.) The line then represents the 
median, but the exact magnitude is not shown in 
figures. 

To represent the median graphically on a surface of 
frequency, find the point on the base line of the surface 
of frequency represented by the calculated median and 
through it erect a perpendicular. This perpendicular is 
sometimes erroneously called the median, but really the 
median is the size of the magnitude on the horizontal 
scale at the foot of the perpendicular. (See Figure 20.) 
This perpendicular divides the surface of frequency into 
two equal parts. 

Through the point on the base representing the median (11.98) 
draw a perpendicular (represented by a dotted line here) which will 
cut the surface into two equal areas. The small squares may be 
counted to prove this. There are 88 small squares to the left of the 
11-11.99 group and 109 to the right of it. The 11-11.99 group 
contains 22 small squares, of which .98 must go with the ones to the 
left, and .02 with the ones to the right. .98 of 22 = 21.56. Adding 
to 88 we get 109.56. .02 of 22 = .44. Adding to 109, we get 109.44, 
or approximately the same area. The slight discrepancy is due to 
carrying the decimals out to only two places. 

Advantages of the Median for School Work. 1. The 

median can usually be located exactly without much 
trouble. 

This is of great service where the mode cannot be exactly deter- 
mined, as in distributions where practically any legitimate groupings 
will give several groups of about the same size. 



Measures of Type 139 

2. Extreme cases influence it little. In this it resembles 
the mode. 

For example, dropping off a number of cases at either end of the 
distribution table on page 134 would only shift the median a part of 
one step. As long as the median stays within the 22 cases of step 11, 
for instance, one case dropped at either end could shift the median 
only jV or .045 of one step. 

3. Its location can never depend upon a small number 
of items. 

It falls at the midpoint of the distribution irrespective of how 
many cases are in a group or where the groups are. 

4. If the number of extreme cases is known, or known 
approximately, we do not have to know their size. 

Thus, if we wish to know the median or typical salary of Latin 
teachers in a state, it is not necessary to get the salaries of teachers in 
the large high schools that will not report to the state superintendent, 
nor in the small unorganized high schools, provided we know about 
how many teachers are in each class of schools. If we know these 
numbers approximately, we can still get the median salary of Latin 
teachers for the whole state. 

5. The median is of special value for data where the 
items cannot be measured in definite units. 

Thus, we may get the median child on any particular ability for a 
room, or the median performer in a debating or oratorical contest, 
without ever being able to measure in definite units the performance 
of a single contestant. An arithmetical average cannot be calculated 
here with any useful accuracy, but the median can be found and com- 
pared with similar medians. 

Disadvantages of the Median for School Statistics. 
1. The median is not so easily calculated as the average. 

The average is computed by an arithmetical process familiar to 
most children in the grammar grades, and it may be computed without 



140 School Statistics and Publicity 



1 



rearranging the items. The median cannot be calculated until the 
data are rearranged in order of size, and while the process of calcula- 
tion is simple, the previous examples show that considerable care 
must be exercised in getting it. 

2. The total cannot be gotten by multiplying the 
median by the number of items. In this respect it is 
like the mode. 

3. It is not useful in those cases where it is desirable 
to give large weight to extreme variations. 

Thus, the average daily attendance of a school is much affected by 
those who are absent a large part of the time. The median attendance 
would not be so affected. But what we wish in this particular in- 
stance is that the great effect of the few extreme cases shall exert its 
full influence. Consequently, the average daily attendance is used. 

4. Unlike the mode but like the average, the median 
may be located in the distribution where the actual cases 
are few. 

5. In a discrete series there may be so many cases the 
same size as the median that it will become almost 
meaningless. 

It cannot mean much here unless there is some reasonable basis 
for regarding the given measures as spread out, that is, for regarding 
the distribution as in some ways continuous. 

EXERCISES 

1. Calculate the median from both ends for each of the distribu- 
tion tables used in previous exercises. 

2. Draw the line to represent the position of the median in each 
of the surfaces of frequency used in previous exercises. 



f 



Measures of Type 141 

III. THE AVERAGE 

Definition. The average is a measure much used in 
ordinary life without being defined. Indeed, it is not 
capable of easy definition. The ordinary person, if 
pinned down long enough, will define it as a measure 
which will give '* the general run " of a group by taking 
into account both the number of cases and the size of 
each one. 

But actually he means one of two things, which differ 
widely from each other and neither of which really 
corresponds to this definition. Most of the time the 
" average " means to him the most frequent measure in 
the group, i.e., the mode. But in some instances, it 
means to him a very unreal and unjust thing obtained 
by statistical sleight of hand. Thus he will say that 
there is no such thing as an average boy, nobody ever 
saw such a boy, etc. This comes nearer his definition 
than does his other use of the term. But even here he 
probably is not conscious of the fact that the average 
is really the size of the balancing point or center of 
gravity in the distribution. 

Calculation. 1. Ordinary method. Ordinarily, the 
average is calculated by dividing the sum of all the 
measures (or cases) by the number of measures. By 
formula it is : 

. /AN Sum of all measures 

Average (Av.) = — ;jr^ 5 

No. of measures 

Thus, for the results on the Courtis Tests, page 134, we could find 
the average as follows : 



142 School Statistics and Publicity 



No. 






• 




Problems 




No. 




All 


Attempted 




Attempting 




Measures 


24 


X 


5 


= 


120 


23 


X 


2 


= 


46 


22 


X 


2 


= 


44 


21 


X 


1 


= 


21 


20 


X 


5 


= 


100 


19 


X 


5 


= 


95 


18 


X 


5 


= 


90 


17 


X 


7 


= 


119 


16 


X 


13 


= 


208 


15 


X 


15 


= 


225 


14 


X 


10 


= 


140 


13 


X 


15 


= 


195 


12 


X 


24 


= 


288 


11 


X 


22 


= 


242 


10 


X 


26 


= 


260 


9 


X 


21 


= 


189 


8 


X 


19 


= 


152 


7 


X 


9 


= 


63 


6 


X 


8 


= 


48 


5 


X 


3 


= 


15 


4 


X 


1 


= 


4 


3 


X 


1 
219 


2] 


3 

.9)2667 




12.18 Average No. 










Problems Attempted. 



2. Short method. By experienced workers in statistics 
averages are often computed by the following method of 
guessing the average and then correcting it. 

First arrange figures in a distribution table. Then 
guess the average by inspection, usually taking the 
approximate median. To be absolutely correct, this 
must be guessed within one step of the true average. 

Then correct the guessed average by the average of all 
the deviations from it. 



Measures of Type 



143 



In the Courtis Test problem just worked, the procedure is as fol- 
lows: 

Guessed average is 11. 

Deviations above or + deviations are : 





+ 


+ 


24 deviations of 


1 or 


24 


15 


2 . 


30 


10 


3 . 


30 


15 


4 . 


60 


13 


5 . 


65 


7 


6 . 


42 


5 


7 . 


35 


5 


8 . 


40 


5 


9 . 


45 


1 


10 . 


10 


2 


11 . 


22 


2 


12 . 


24 


5 


13 . 


65 



+492 



Deviations below or — deviations are : 



26 deviations of 1 or 26 



21 ... . 


. 2 . 


. 42 


19 ... . 


. 3 . 


. 57 


9 . . . . 


. 4 . 


. 36 


8 . . . . 


. 5 . 


. 40 


3 . . . . 


. 6 . 


. 18 


r-l 


. 7 . 


. 7 


1 . . . . 


. 8 . 


. 8 
-234 


+ Deviations 492 






234 






Excess of + deviations =258 







This means that we got the guessed average too low because there 
are more deviations above than below it. The 258 is the excess of 
deviations above. As there are 219 cases, the average excess devia- 
tion to satisfy every case is |f| or 1.18 of a step. As the guessed aver- 



144 School Statistics and Publicity 

age of 11 was too low, we add this 1.18 average deviation and get' 
12.18 for our corrected average. 

Had the minus deviations been in excess, it would have meant 
that the guessed average was too high, and we should have sub- 
tracted the correction. 

The rule for computing the average by the short-cut method 
is: " Arrange the numbers in the order of their magni- 
tude ; choose any number hkely to be nearest the average ; 
add together, regarding signs, the deviations from it of all 
the numbers; divide this result by the number of the 
measures of the average which you are obtaining; add 
the quotient to the chosen number." ^ 

In using the short-cut method, three cautions must be 
kept in mind : 

1. To get much accuracy, the guessed average must be within one 
step of the true average. 

2. The correction must be added to the guessed average if the 
plus deviations are in the majority; it must be subtracted if the 
minus deviations are in the majority. 

3. If the guessed average is in a group, the deviations for all that 
group are 0. The number of cases, however, must be used in dividing 
to get the average deviation. 

The short method is not always short by itself ; but it 
often saves time on certain calculations on deviations. 

Graphic Representation. The average can be rep- 
resented graphically in the same way as the median. 
(See page 137.) That is, the point representing the 
calculated average is found on the base line and a per- 
pendicular is erected to call attention to the size of the 
average. However, there is in general little value in 
representing the average this way because the line drawn 
has no appreciable relation to the surface of frequency. 

1 Thorndike, E. L. : Mental and Social Measurements, p. 3 



Measures of Type 145 

A line for the mode runs through the highest part of 
the surface, a fact the eye easily grasps. A line for the 
median cuts the surface into two equal areas, which the 
eye will readily compare. But the average has no such 
relationship to show. 

Advantages of the Average for School Statistics. 
1. " Unlike the median or mode, it may be definitely 
located by a simple process of addition and division, and 
it is unnecessary to draw diagrams or arrange the data in 
any set form or series." ^ 

The great ease with which the average can be calculated from 
figures in almost any form is doubtless the main reason it has been so 
often used. It is not even necessary to throw figures into a distribu- 
tion table or table of frequency, as is the case in finding the median 
or the mode. 

2. It weighs extreme cases, which is a desirable thing 
in certain instances. 

3. " Unlike the mode, it is affected by every item in the 
group, and its location can never be due to a small class 
of items." 2 

Thus Superintendent Spaulding in a recent bulletin gets the aver- 
age expenditure for Minneapolis by taking the average for five years. 
No one year is any more important than another and so gets counted 
no more and no less than another year. The median in such a small 
number of cases, with an odd number especially, would emphasize 
one year unduly. 

4. The method of calculating it is familiar to every one. 

5. It may be determined when the aggregate and the 
number of items are the only things known. 

Thus, in determining the typical number of days' attendance for 
a child in a certain grade, we may have the total number of days' 

1 King, W. I. : Elements of Statistical Method, p. 136 

2 Ibid., p. 136 



146 School Statistics and Publicity 



■ 



attendance and the total number of children. We might not have the 
actual record of a single child given us. But from the other two items 
we could calculate the average number of days' attendance for a child. 
Sometimes these are the only two items that can be secured from 
reports. Thus we may have only the total for teachers' salaries and 
the number of teachers ; the total sum spent on repairs and the num- 
ber of buildings; the total expenditures for a certain kind of school 
and the number of such schools, etc. 

Disadvantages of the Average for School Statistics. 

1. It cannot be located on a surface of frequency from 
looking at the surface alone. 

The median or mode can be quickly located by inspecting the sur- 
face of frequency. The average can be put on it only after the aver- 
age has been calculated, and the point counted out on the base scale. 

2. It cannot be located accurately if the extremes are 
missing or in any way doubtful. 

This is particularly troublesome when we recall that we are often 
inclined to expect extreme cases of expenditures and such things 
in school work to be doubtful or inaccurate. 

3. It lays too much stress on extreme variations. 

This is as bad a thing for school statistics as judging a patent 
medicine by the enthusiastic few who write testimonials about it for 
publication, or a school by what is done the first and last week in each 
session. The trouble can, however, be eliminated to some extent 
by dropping very extreme cases and calculating the average from the 
remaining ones. 

4. It cannot be used where we cannot accurately 
measure the quantities studied. 

The so-called averaging of points on contestants in a debate is often 
futile because they are not measured by the different judges on the 
same scale. 

5. It may fall where no data actually exist. 

That is, it may fall in a gap (the median does this sometimes as 
well) and be almost as absurd as the case of the duck reported by a 



Measures of Type 147 

western professor. The duck was shot at with a double-barreled 
shotgun, one shot going two feet to the right of the duck and the other 
two feet to the left. The average performance was, of course, zero 
and centered on the duck. Statistically the duck was dead ; actually 
it flew away. 

6. The average often means to the ordinary man 
something different from what the calculated thing means. 

The process of calculation is undoubtedly familiar to all. But 
what the ordinary man often means by the "average boy" is a boy 
like the majority of boys, not like the few above or below this majority. 
The man in all probability has seldom thought through the fact that 
his method of calculation does not always give him this "majority" 
measure. 

EXERCISES 

1. Calculate by both the long and short method the average for 
each of the distribution tables used in previous exercises. 

2. Draw the line to represent the position of the average in each 
of the surfaces of frequency used in previous exercises. 

IV. WHICH MEASURE OF TYPE TO USE IN A GIVEN 

DISTRIBUTION 

If the distribution is symmetrical or approximately so, 
the average, median, or mode are, of course, the same or 
approximately so. Consequently, as regards magnitude, 
it is a matter of indifference which of the three is used as 
a measure of central tendency. But it is generally im- 
possible to tell whether the distribution is symmetrical 
or not till a frequency table has been made. Once this 
table has been made, it is much easier to determine the 
median than the average. In a Bobbitt table or similar 
distribution where all cases are separate, the median 
can be found much more quickly than the average. 

The mode is the measure to use in all skew distributions 
or bi-modal ones. 



148 School Statistics and Publicity 

The matter of symmetry or skewness can be seen at a 
glance from the surface of frequency. With a little 
experience it can be recognized directly from the table of 
frequency. 

For school statistics, this is probably the safest rule : 

Use the mode for skew and hi-modal distributions; 
the average, for cases where every item must he counted 
as much as any other item {say in finding the average 
expenditures for five or less years); and the median for 
all others. 

Of the three, the median is by far the best " all pur- 
pose " measure of central tendency for school statistics. 

EXERCISE 

In each of the distribution tables for which you have calculated 
the mode, median, and average, which is the best measure of central 
tendency ? Why ? 

REFERENCES FOR SUPPLEMENTARY READING 

King, W. I. Elements of Statistical Method, Chapter XII. 

Rugg, H. 0. Statistical Methods Applied to Education, Chapter V. 

Thorndike, E. L. Mental and Social Measurements, pp. 36-39. 



CHAPTER VI 

MEASURES OF DEVIATION OR DISPERSION 

The second element necessary in giving the bird's-eye 
view, it will be recalled, is some measure of how much 
the cases range from the central tendency. This is 
variously called '' spread," " range," " deviation," or 
" dispersion." It is of just as much importance as the 
central tendency. For example, it makes a vast difference 
whether a teacher is called upon to teach children that 
are all very close to the typical or " average " child, or 
ones that vary enormously both above and below. 
Supervision means one thing for a superintendent when 
all his teachers are very close in ability, experience, 
spirit, etc., to his typical teacher; it means an entirely 
different thing when many of his teachers range widely 
either side of this typical teacher. 

There are several ways of measuring this range, the 
most useful of which will now be given. 

I. EXTREME RANGE VARIATION 

The dispersion may be shown by giving the extreme 
cases and thus showing how far it is between them. This 
measure may be useful in showing the difference of ability 
in achievements of a grade in school. 

For example, the eighth grade in a western city as reported on 
some Courtis Tests varied in attempts on the addition problems 
from 4 to 24, or a range of 20 out of a possible range of 24. This 

149 



150 



School Statistics and Publicity 



measure shows at once that there is a wide range of ability of eighth 
grade children in that city as regards attempts on addition problems. 

But as a general rule, the extreme range variation is not 
a good measure of dispersion because the extreme cases 
are likely to be very unreliable, especially if there are 
only one or two that are isolated from the rest of the 
measures in the distribution. 

The range of teachers' salaries in the typical school system is 
not from that of the superintendent down to the salary of the poorest 
paid teacher, because there is only one superintendent and his salary 
is far removed on the scale from that of the highest paid teacher under 
him. His salary must be eliminated before the extreme range for 
the others has any value. 

The influence of extreme variations upon this measure is shown 
by Table 17, taken from a study in costs of instruction in home 
economics in fifteen southern normal schools.^ 

Table 17. Variations in Cost of Instruction in Home 
Economics in 15 Southern Normal Schools, per 1000 Stu- 
dent Hours, 1916 



Key number of 

normal school 

III 

VII 

XIII 

II 

VIII 

IV 

XI 

XIV 

XV 

I 

VI 

X 

V 

IX 

XII 



Cost of home economics 

instruction per 1000 

student hours 

$194 

115 

89 

88 

85 

82 

79 

73 

70 

64 

60 

58 

53 

52 

15 



1 Alexander, Carter: "Costs of Instruction in Normal Schools," 
Elementary School Journal, XVII, 653 



Measures of Deviation or Dispersion 151 

The extreme range variation between 194 and 15 is 179. But it 
is very evident that both extremes are the results of some very unusual 
factor. If the top case is cut off, the range is reduced to 100. If the 
bottom case is cut off, but the top one retained, the range is 142. 
If both are cut off, the range is reduced to 63. If the two top cases 
and the bottom one, all of which seem unusual, are eliminated, the 
range is reduced to 37, which is probably a fairly reliable figure. 

The uncertainty as to just how many cases to leave 
out in any one distribution when computing the spread 
emphasizes the need of having some standard proportion 
cut off, say a fourth at each end. Then the average 
deviation or spread from the central tendency of the two 
intersections or points thus obtained can be computed. 
Let us now discuss some of the devices for getting devia- 
tions from such points. 

II. QUARTILE DEVIATION (SEMI-INTER-QUARTILE RANGE 

OR " Q ") 

This is by far the easiest and most important measure 
of dispersion for school statistics. But before it can be 
easily understood, it is necessary to get the quartiles. 
These are the magnitudes of the points on the scale which 
divide the distribution into four equal groups of cases (or 
quarters) . 

Quartiles. Obviously, it takes three such points to 
divide the whole into four groups with equal numbers of 
cases, — the first quartile or 25 percentile, the second 
quartile (which, of course, is the median), and the third 
quartile or 75 percentile. Obviously, also, the first 
quartile is the median of the lower half of the distribution, 
and the third quartile is the median of the upper half. 
Below the first quartile come one-fourth of the cases 
and above it, three-fourths. Above the third quartile 



152 School Statistics and Publicity 

come one-fourth of the cases, and below it, three-fourths. 
Between the first and third quartiles come one-half or 
50 per cent of the cases. 

The problem of locating the first quartile is statistically 
the same as that of locating the median, except that cases 
are counted in from one end so as to get only one-fourth 
of them on one side. That is, the object this time is to 
find the " quarter point " from one end of the distribution. 

In a Bobbitt tableit is generally best to fix the number of cases so 
as to know in advance whether the quartiles will be shown as separate 
cases or fall between separate cases. In this way, there will be no 
trouble with splitting cases. Thus if the number of cases is some 
multiple of 4, plus 1, the median will appear as a separate case and 
the quartiles will fall between cases., For the Bobbitt table on page 
18, the quartiles are .respectively 47 for Q 1 (|^ of 42 + 52) and 78 for 
Q 3 (I of 74 + 82). If the total number of cases is some multiple of 
4, the median and both quartiles will fall between cases. If the total 
number of cases is some multiple of 4, plus 3, the median and both 
quartiles will appear as separate cases. 

The calculation of the quartiles in the distribution given on page 
134 is as follows. Since there are 219 cases, the first quartile must 
come at the point that will throw | of 219 or 54 1 cases below it. 
Counting up, we find that this will require 13 f cases of the 9-10 group, 

there being only 41 cases below. This will make it come -^ — or 

21 
.65 of the step up. Q 1 then is 9 + .65 = 9.65. Similarly Q 3 is 

m the 15 group commg down. 54| — 45 = 9|. ^^-^ = .65. 

Q 3 then is 16 - .65 = 15.35. ^ 

Graphically, the first quartile is the magnitude on the 
base at the foot of a perpendicular so located that it will 
cut off one-fourth of the area of the surface of frequency 
to the left of it and three-fourths to the right. The third 
quartile has one-fourth of the area to the right and three- 
fourths to the left. The median has one-half the area 
on either side of it. (See Figure 20, page 137.) 



Measures of Deviation or Dispersion 153 

Quartile Deviation. The Q is found by taking half 
the difference between the first and third quartiles. This 
average will give the average distance of the quartiles 
from the median of central tendency selected. The aver- 
age distance is used because sometimes the median is not 
exactly halfway between the quartiles (it cannot be if the 
distribution is not perfectly symmetrical). 

By formula, then, 

^ Quartile 3 — Quartile 1 
Q = 2 

For the Bobbitt table on page 18, the calculation is : 

Q = $Z§-iiML=i?l= $15.50 

For the distribution on page 134, the calculation is: 
_ 15.35 - 9.65 ^ 5.70 ^ 2 85 

Li Li 

Graphically, Q is one-half the distance from Quartile 3 
to Quartile 1 on the base of the surface of frequency. But 
as it is difficult to show only one-half of this, the whole of 
this distance is usually represented as 2 Q . (See Figure 20 . ) 

III. OTHER PERCENTILE DEVIATIONS 

In some of the educational investigations, especially 
those issued by workers at the University of Chicago, 
dispersion is indicated by the distance from the median 
of some point marking off a convenient number of the 
cases, other than the quartile. The number of cases is 
usually a common percentage; hence the point is called 
a *' percentile." In some studies the cases are divided 
into thirds by " tertiles." The median here is, of course, 
in the middle of the second group. The tertile deviation, 
then, is half the difference of the first and second tertiles 



154 



School Statistics and Publicity 



(only two tertiles are needed to make three groups). 
Similarly four '' quintiles " will divide the distribution 
into five groups with the same number of cases, the median 
being at the middle of the third group. The quintile 
deviation then would be one-half the difference of the 
first and fourth quintiles. 

Table 18. Table to Show Variation by Percentage Groups, 
Using Distribution of Annual Salaries of Regular Teach- 
ers IN Elementary Schools in Cleveland and in 13 Other 
Cities of More Than 250,000 Inhabitants ^ 





Salaries not exceeding the amounts specified were 




earned by teachers bearing to the aggregate num- 


City 


ber employed in each city the proportion of: 




10 per 


30 per 


50 per 


70 per 


90 per 




cent 


cent 


cent 


cent 


cent 


Baltimore , . . 


$600 


$ 700 


$ 700 


$ 750 


$ 800 


Boston .... 


648 


840 


1,176 


1,176 


1,224 


Chicago . . . 


675 


975 


1,175 


1,175 


1,200 


Cincinnati . . . 


700 


900 


1,000 


1,000 


1,000 


Cleveland . . . 


600 


750 


900 


1,000 


1,000 


Indianapolis . . 


475 


625 


875 


925 


925 


Milwaukee . . 


876 


876 


876 


876 


876 


Minneapolis . . 


750 


950 


1,000 


1,000 


1,000 


Newark .... 


630 


780 


1,000 


1,000 


1,300 


New Orleans . . 


500 


600 


700 


750 


800 


Philadelphia . . 


630 


780 


900 


940 


1,000 


San Francisco . . 


840 


1,164 


1,200 


1,224 


1,224 


St. Louis . . . 


700 


1,032 


1,032 


1,032 


1,120 


Washington . . 


625 


700 


750 


890 


980 


Average . . . 


$661 


$ 834 


$ 949 


$ 988 


$1,032 



^ Data for Cleveland from payroll for 1914-15; data for other 
cities for 1913-14, from "Tangible Rewards of Teaching," U. S, 
Bureau of Education 



Measures of Deviation or Dispersion 155 

The calculations and graphic representations for these 
various percentile deviations are precisely similar to those 
for the Q or quartile deviation. 

These deviations are found as yet in few educational 
investigations, but should be understood for reading 
purposes. 

A simple but effective device for popular consumption 
indicates the variability indirectly by showing the magni- 
tudes which equal or exceed certain fixed percentages of 
the cases. This device has been found especially valu- 
able for making comparisons between different distribu- 
tions. 

A good example occurs in the Cleveland Survey ^ and 
is reproduced in Table 18. 

IV. MEDIAN DEVIATION (MED. DEV. OR P. E.) 

This is the median of all the deviations arranged in 
order of size and irrespective of whether they are above or 
below the central tendency. In other words, it is cal- 
culated by arranging the deviations in order of size and 
then finding the median of these deviations in precisely 
the same way as the median of anything else would be 
found. 

Within the median deviation of the central tendency 
used, the middle 50 per cent of the cases come. Within 
the Q of the central tendency approximately 50 per cent 
of the cases come, but they are not necessarily the middle 
50 per cent, because the central tendency does not 
lie exactly halfway between the quartiles, except in a 
sjnumetrical distribution. In such a distribution the 
median deviation is of course the same as Q. 

1 Volume on ** Financing the Schools," p. 57 



156 School Statistics and Publicity 

The median deviation is little used in school statistics, 
but the superintendent may come across it in his reading 
of educational investigations and in attempts to indicate 
variability. It is sometimes called the '' probable error," 
and this is where it gets the letters '* P. E." However, 
this name is not a good one for, as Professor Thorndike 
suggests, the median deviation '' is not specially probable 
and not an error at all." ^ A statement of central 
tendency of 30 with a P. E. of 4 means that half the cases 
deviate more than 4 and half of them less than 4 from this 
30. It also means that for any given case chosen at 
random, the chances are 50 to 50 that it will deviate more 
than 4 from this 30, that is, be below 26 or above 34; 
the chances are likewise 50 to 50 that it will deviate less 
than 4 from this 30, that is, be between 26 and 34. In 
other words, it is a toss-up as to whether any given case 
will be likely to deviate more or less than 4 from this 
central tendency of 30 ; that is, any given case is as likely 
to be between 26 and 34 as it is to be below 26 or 
above 34. 

V. AVERAGE DEVIATION (A. D.) 

This is simply the average of all the deviations from 
whatever central tendency is selected. It may be figured 
from any measure of central tendency. But it is best 
calculated from the average or approximate average, 
next from the median or approximate median, and seldom 
if ever from the mode, unless the average spread for each 
side is given separately. 

The A. D. for the Bobbitt table on page 18 is figured 
thus : 

» Mental and Social Measurements, p. 40, footnote 



Measures of Deviation or Dispersion 157 

Deviation from median (59) of 61 is 2 



62 


3 


63 


4 


69 


10 


69 


10 


74 


15 


82 


23 


88 


29 


100 


41 


100 


41 


112 


53 


169 


110 


59 





58 


1 


56 


3 


56 


3 


54 


5 


53 


6 


52 


7 


42 


17 


41 


18 


38 


21 


34 


25 


33 


26 


30 


29 


!es, 25)502 sum 




20.1 A. D 



The A. D. for a distribution in groups is usually figured 
from the approximate average or the approximate median, 
so as to have the deviations in whole steps. The result 
is close enough for all practical purposes. 

The A. D. for the distribution given on page 134 is figured thus 
from the guessed average : 

492 + deviations ] ^ . ^ 

234- deviations l^^^P^S^ 143 

No. of cases, 219 )726 sum of all deviations 

3.31 A. D. 



158 School Statistics and Publicity 

VI. STANDARD DEVIATION (MEAN SQUARE DEVIATION, S. D.) 

This is found by taking the square root of the average 
of the squares of the deviations, counting zero deviations. 

^^ -p. _ / Sum of squares of deviations 
^ Number of deviations 

It is of no particular value in school statistics except 
that the superintendent should understand it so that he 
may read intelligently educational or economic treat- 
ments that use it. It saves time to use this as the measure 
of dispersion if the Pearson Coefficient of Correlation is 
later to be used. It is better than the A. D. if the de- 
viations of the extreme cases need to be weighted. 

The calculation of the S. D. from the guessed average in the dis- 
tribution on page 134 is figured as follows, using the deviations as 
given on page 143 : 



24 X 12 


= 


24 


15 X 22 


= 


60 


10 X 32 


= 


90 


15 X 42 


= 


240 


13 X 52 


= 


325 


7 X 62 


= 


252 


5 X 72 


= 


245 


5 X 82 


= 


320 


5 X 92 


= 


405 


1 X 102 


= 


100 


2 X 11^ 


= 


242 


2 X 122 


= 


288 


5 X 132 


=: 


845 


26 X 12 


= 


26 


21 X 22 


= 


84 


19 X 32 


= 


171 


9 X 42 


= 


144 


8 X 52 


= 


200 


3 X 62 


= 


108 


1 X 72 


= 


49 


1 X 82 


= 


64 



^^ 



S.D. =j4-282=4.42 
219 



4282 



Measures of Deviation or Dispersion 159 

VII. DEVIATIONS FOR SKEW DISTRIBUTIONS 

The measures for deviations so far given are serviceable 
only for symmetrical, or approximately symmetrical, 
distributions. If the distribution is a skew one, it is 
evident that no average deviation for the whole can be 
of any service, because the deviation on one side of the 
central tendency will be markedly different from the 
deviation on the other side. This average or median 
deviation for the whole would be a deviation which did 
not actually exist. 

Hence in a skew distribution it is customary to take 
the mode or median (preferably the mode) for the central 
tendency, and then to give the average or median deviation 
for the cases above this central tendency, and the same 
for the cases below it. 

This procedure may be illustrated from the salaries paid the 
Cleveland elementary school teachers in 1914-15.1 (See page 160.) 

The approximate median is sufficient here and inasmuch as only 
an average deviation is to be figured, it is unnecessary to make the 
groupings by even steps. 

There are in all 2204 teachers, making 1102 above the median and 
1102 below. The A. D. above is • 

$118810 (MAO 

The A. D. below is 

$200920 »ia, . 
1101"^^^^^ + 
The median of the part above the group containing the median 
is at the end of the 525th case, evidently in the $1000 group, making 
the quartile deviation above roughly $100. The median of the part 
below the group containing the median is evidently in the $700 
group, making the quartile deviation below roughly $200. The 
exact median and the exact quartile deviation could be figured if 
desired, counting 1102 cases above the median and 551 cases above 
the quartile. 

1 Adapted from Cleveland Survey, Summary Volume, p. 98 



160 



School Statistics and Publicity 



Salary paid 


Number getting 
salary 


Deviation from 
median 


Sum of 
deviations 




$1650 


1 


750 


+ 750 




1540 


2 


640 


1280 




1500 


4 


600 


2400 




1430 


1 


530 


530 




1400 


1 


500 


500 




1300 


5 


400 


2000 




1210 


1 


310 


310 




1200 


7 


300 


2100 




1155 


1 


255 


255 




1100 


71 


200 


14200 




1050 


83 Total 


150 


12450 




1045 


3 teachers 


145 


435 


Approxi- 


1000 


762 above = 


100 


76200 


mate 


950 


108 1050 


50 



50 


5400 118810 


Median = 


= 900 


196 196 







850 


112 


-5600 




825 


2 


75 


150 




800 


130 


100 


13000 




770 


1 


130 


130 




750 


133 


150 


19950 




715 


4 


185 


740 


* 


700 


164 


200 


32800 




650 


145 


250 


36250 




600 


136 Total 


300 


40800 




550 


20 teachers 


350 


7000 




500 


110 below - 


400 


44000 




400 


1 958 

2)2204 

1102 


500 


500 200920 



The deviations have been calculated with reference to the median 
in this particular example because this was the central tendency 
used in the survey. But it is apparent that if the modal salary of 
$1000 as denoted by the largest group of 762 cases were taken, the 
discrepancy between the deviation measures on the two sides would 
be much greater. 



Measures of Deviation or Dispersion 161 



"1200- 
1160 4- 
1 120-1- 
1060- 
1040- 
1000- 
960 r 

920 - 

660 

640- 

eoo- 



Hobak en 



760 -- 



720 -- 



M 



680 

640- 



560- 
520- 
460-U 
440 -- 



Springfield 



Bayonnc 



Youngstown 
Fort Wayne 



D«i Moines 



Passaic -^omervilk— Springfield—Lawrence — New Bedford^Evanay/He-Doiuth 



■ Utica- 



-Canton- 



— Wafer bury f-lolyoke — : Lynn^ 

Pa^ focket^j^^^J^^nton Tcrre ^aute^.^^.^ /Kansas Cy. 

£. St. Louis 

Schenectady Saginaw Sf.Josepii Wi/kes-Barre - 



M , . Harrisbu/^ 
nanchcater 

Altoona Souffj Bend 

Reading 



Fig. 21. — Graphed Bobbitt Table of Mean Annual Salaries Paid Ele- 
mentary Teachers in Certain Cities. 
(J. F. Bobbitt : Elementary School Journal, 15 : 45.) 



162 School Statistics and Publicity 

In this connection it should be noticed that a Bobbitt 
table may be really a skew distribution. This skewness 
will not be apparent at a glance because the table will 
show just as many lines or cases between one quartile 
and the median as between the other quartile and the 
median. It needs inspection to see whether cases listed 
on separate lines are really the same in size or approxi- 
mately so. Professor Bobbitt removes this difficulty by 
graphing his results. (See page 101.) The crowding in 
of the cases below the median shows that the devia- 
tion is on the average much less on that side than on 
the upper. The same is true of the graph given in Fig- 
ure 21. 

VIII. WHICH MEASURE OF DEVIATION TO USE IN A GIVEN 

DISTRIBUTION 

The choice of the measure of deviation depends upon 
the measure of central tendency selected. If the latter is 
the average, then the deviation should be expressed by 
the average deviation. If the median is used, deviation 
should be expressed by the Q, although the average de- 
viation may be employed. If for any reason extreme 
variations are to be emphasized, the extreme range 
measure or the standard deviation should be employed. 
In a skew distribution, if the median is used, the quartile 
deviation or the average deviation for each side should be 
given separately; if the mode is used, the average 
deviation or the median deviation for each side should be 
given separately. 



Measures of Deviation or Dispersion 163 

EXERCISES 

1. Calculate all the measures of deviation or dispersion you can 
for each of several of the distribution tables used in previous exercises. 
For this use the measures of central tendency chosen in the exercise 
on page 148. 

2. For each of the distributions used in the exercise above, which 
measures of deviation are preferable? Why? 

REFERENCES FOR SUPPLEMENTARY READING 

King, W. I. Elements of Statistical Method, Chapter XIII. 

Rugg, H. O. Statistical Methods Applied to Education, pp. 149- 

173 and 178-180. 
Thorndike, E. L. Mental and Social Measurements, pp. 46-50 and 

Chapter VI. 



CHAPTER VII 

MEASURES OF RELATIONSHIPS 

So far we have for the most part considered distribu- 
tions of measures as wholes and a single distribution at 
a time. But one of the greatest values of statistical method 
is the ease with which it makes possible the study of 
relationships, including those between separate distribu- 
tions. Since things have no meaning except through 
their connections with other things, statistical method 
must bring out such relationships very clearly. It was, 
of course, impossible to cover the previous topics in 
this book without dealing with relationships. But it 
is now advisable to give these connections a special and 
separate treatment. 

I. RELATIONSHIPS INSIDE OF ONE GROUP 

Discrete Series. As soon as a Bobbitt table or cen- 
tral tendency and the deviation for any distribution 
are given, by any of the various ways for calculating 
these measures, a relationship is, of course, indicated. 
But in general, inside of one distribution, relationship is 
most easily shown by some form of the bar graph. 

If every item is kept separate, each should be repre- 
sented by a separate bar. All bars should be the same 
width and differ only in length. This length indicates 
the measure of the case. 

164 



Measures of Relationships 165 

The graphing of the Bobbitt table in Figures 8 and 9 is an example 
of this. The bars in Figure 8, of course, may run vertically if pre- 
ferred. 

In such a table the cases are arranged from high to low, and the 
tops of the bars or the curve for the graphic presentation will have a 
general downward or upward direction, according to the end from which 
it is viewed. But there are some distributions in which the cases come 
in a time sequence or in an alphabetical order so that the tops of the 
bars or the curve jog up and down. This would be the case with the 
number of children in school by years or months in a city of fluctuat- 
ing population, if the base scale represented the calendar months or 
years in succession. In this instance, the central tendency and meas- 
ure of deviation would have to be calculated and given separately, 
or else shown with the same data rearranged as a Bobbitt table or 
ordinary grouped distribution. 

Continuous Series. If there is more than one case to 
an item, that is, if the items are grouped, the length of 
each bar will represent the number of cases. This 
amounts then merely to drawing the surface of frequency 
with bars representing groups which may or may not be 
adjacent to each other. ^ The tops of the ends of these 
bars form the broken line or ''curve" which, united with 
the base line, makes up the surface of frequency. 

If the distribution table is to be written or printed 
without graphing, the relative sizes of the groups may 
frequently be brought out more forcibly by turning the 
numbers into percentages of the whole number of cases. 
Of course, the ratio between 47 cases of magnitude 6 in a 
total of 470 cases is exactly the same as the ratio of 10 
per cent to 100 per cent. But the ordinary man is used 
to thinking in terms of percentages and can grasp the 
relationship much more quickly that way. Graphing 
will show this relationship as easily from the original 
figiires as from percentages, so there is no need for the 

1 See p. Ill 



166 



School Statistics and Publicity 



change to percentages where graphing is to be done, in 
the case of one distribution taken by itself. 

II. SIMPLE RELATIONSHIPS BETWEEN DIFFERENT DIS- 
TRIBUTIONS 



If two different distribu- 
tions have been made up on 
the same scale, they can be 
conveniently shown in bar 
graphs, as in Figure 22. 

It may be noted there 
that the tops of the two dif- 
ferent sets of bars give two 
" curves." This device will 
not be of service in compar- 
ing more than two distribu- 
tions at a time, but several 
more " curves " may be 
drawn provided they do not 
overlap too much. It is cus- 
tomary to represent curves 
alone by different conven- 
tional signs as : 

No. IV 

No. V -.-.-.-.-.-.-.-.-.-.-. 

No. Yl fill II II 11 II II 1 1 II II II II II II II II II 




Fig. 22. — Device for Comparing 
Two Different Distributions Made 
up on the Same Scale with Bar 
Graphs. 

The whole bara represent the total en- 
rollment, while the shaded portions repre- 
sent retarded children in the grades at 
Memphis, about 1908. (From Laggards in 
Our Schools by L. P. AjTes, page 39, by 
permission of Russell SageFoundation) 

No. I 

No. II - 

No. Ill rrrrrr....^.^ 



Figure 23 is a good example. 

However, it is often very difficult to grasp comparisons 
between two different distributions using the same scale, 
unless the number of cases for each magnitude is put 
on the same basis as regards the rest of its distribution. 



Measures of Relationships 



167 



RESULTS OF SPELLING TESTS 
PERCENTAGE OF WORDS SPELLED COR- 
RECTLY BY GRADES 

Percent 
100 



90 



80 



70 L 



60 



Highest 1 
Average UuTTE 

Lowest J 
Average for 
schools in 22 
cities 



This is most easily done by changing the number of cases 
in each step to the proper percentage of the whole. That 
is, one would not give the achievements in composition 
for several school grades 
by merely citing the 
number of children in 
each grade making the 
various scores. Instead, 
he would give tables 
presenting the percent- 
ages of children making 
each score in each grade. 

Table 18 on page 154 
employs percentages for 
making comparisons be- 
tween cities on salaries 
of teachers, by groups. 

A good way to compare 
several distributions is to 
place the surfaces of fre- 
quency made up from 
the percentage tables one 
above the other. This, 
of course, may be used 
only when they are all 
applied to the same 
scales. In this case, 
they must be centered 
exactly (the centers in 
the same vertical line), or else the differences between the 
central tendencies must be drawn to scale accurately. 
Figure 24 is a good example. 

Professor Bobbitt in the San Antonio Survey used 



50 



■40 



30 



20 



10 















i 




^ 


^ 




A 


i 


^ 


^ 


1 


p 


t 











































































Fig. 



2 3 4 5 6 7 8 Grades 

23. — Use of Curves for Comparing 
More Than Two Distributions. 

This chart represents the range from the 
poorest to the best room tested in each grade 
in spelling, Butte, Montana. For example, 
the poorest second-grade room averaged 73. 
The average for the whole city is repre- 
sented by the dotted line, while the average 
for twenty-two cities is represented by the 
heavy line at 70. (From Butte, Montana, Sur- 
vey, page 72) 



Scores 



2 3 4 5 6 789 

Median Scores 




Scores: 01 23456789 



Fig. 24. — Graph for Comparing Related Distributions, Using the Com- 
position Scores in Salt Lake City. 

(From Salt Lake City Survey, page 141) 



168 



SPELLING ABILITY BY GRADES 



100 

eo 

QO 
70 
60 

Q^ 

50 

M 
AO 

Or 

30 
10 



Cira de W 



3 26 

14 



19 15 

3 



27 



7 



^^4 ,2^ 

17 
2? 6 ?4 



8 



n 



Q, 



GrnrioW 



19 



15 



S 3 

13 7 

20 (5 J8 



% /^ 2 ^ '4 



22 
24 



33 
9 



-22. 



-21- 



Gmdfl 7 



/6 



Ji. 



"IW24 T2 

4 _ 5 :2r 



^4. 



Fig. 25. — Device for Graphing Bobbitt Tables from Different Distribu- 
tions on Related Scales. 

Each wide column represents the achievement in spelling of the ward schools for 
that grade, on the Ayres scale. Each number represents a ward school, and the height 
on the scale shows the achievement of that grade in that school. The quartile and 
median lines are shown. (Adapted from the San Antonio Survey, page 105) 

169 



170 School Statistics and Publicity 

another device for making comparisons between different 
distributions that had been made up as Bobbitt tables. 
He drew their quartiles and medians as horizontal lines 
across vertical columns, allowing each distribution one 
column and adjusting the horizontal lines on the vertical 
ones as scales. Figure 25 gives part of his device for 
showing achievements of different grades in spelling. 

This is a very convenient graph, for it shows a good 
many relationships at a glance, such as the variations in 
central tendencies, dispersions, etc. By observing only the 
lines belonging to the medians, a curve may be read 
across the page. That is, a little practice will enable 
one to see three curves on the chart, the upper rep- 
resenting quartile 3; the middle, the median; and the 
lower, quartile 1. 

III. COEFFICIENT OF VARIABILITY OR DISPERSION 

A serious difficulty arises when we try to compare 
two distributions as regards the amounts of dispersion 
unless they have about the same central tendencies. If 
the central tendencies are widely different and the abso- 
lute number of units of dispersion is the same, the real 
or relative amounts of dispersion are widely different. 

For example, suppose a superintendent is studying the way his 
high school teachers mark pupils. He gets 500 or more marks given 
by each teacher. He finds that two teachers have the same median 
in their marks, 80 on a numerical scale. But one teacher has a Q of 
5 and the other a Q of 10. In the first case, the typical dispersion is 
6| per cent of the median; in the other it is 12| per cent. Again, 
a median of 90 with a Q of 5 would be markedly different from a 
median of 80 with a Q of 5. In the former, the variability would 
be 5f per cent and in the latter, 6i per cent, although they varied 
10 points in the central tendency. 

Consequently, in comparing distributions or groups 



Measures of Relationships 171 

on variability, experienced persons compare them through 
percentages of variabiUty or dispersion. The percentage 
of dispersion is usually called the coefficient of variability. 
By formula it is simply : 

r^ ra ' ^ £ ^T • i„-T4- Mcasure of Devlatlou . , , 
Coefficient of Variability = — :^ jj^ , pointed 

off as per cent. 

Thus, for the distribution on page 134, the calcula- 
tion is : 

Coefficientof Variability=§ = ^-g (from p 153) ^ 

M 11.98 (from p. 135) 

Professor Haggerty in his study of arithmetic in 
twenty Indiana cities ^ has an interesting graph for com- 
paring variation in two distributions. He had figures 
for each school grade on the Courtis Tests in both attempts 
and rights, thus getting two distributions of twenty cases 
each. So he made a scale for the attempts and one for 
the rights on each grade, with the medians centered and 
opposite. Thus a line joining these two centers was 
horizontal. A given city could be indicated by finding 
its position on each of the two scales and joining these 
points by a line. If the given city varied as did the whole 
group of cities, its line was parallel to the original line 
joining the medians. If it did not vary that way, its 
line would slant, the end farthest from the median line 
indicating that it varied more in that respect than in the 
other. 

Thus in Figure 26, for the fifth grade in addition, the upper heavy 
line represents the work of the city of Bloomington and shows at a 
glance that the fifth grade in Bloomington attempted 9 problems in 
addition and got about 5| of these right ; that in both attempts and 

1 Haggerty, M. E. : "Arithmetic: A Cooperative Study in Edu- 
cational Measurements," Indiana University Studies, No. 27 



14 


■ 




13 




7 


12 i 


■ 




1) 




6 


10 
9 


/ 


B/oomington. 

5 


8 






7 

6 


: 


4 

Median 
2.0 Ind Cities 






3 


5' 






4 










2 


3' 






a 


' 


1 


1 






, 







A R 

Fifth grade addition Bloomington Ind. compared 
with standard for 20 Ind. cities, Haggerty. 

Fig. 26. — Graphic Device for Comparing Variation in Two Different 

Distributions. 

The scale for attempts in the Courtis Tests from twenty cities is shown on the left, 
centered with the scale for rights on the right. The slanting line represents the achieve- 
ment of Bloomington, the slant upward indicating that it is relatively higher in rights 
than in attempts. 

172 



Measures of Relationships 173 

rights it exceeded the standards set by the twenty Indiana cities as 
a group (these standards are 6.5 and 3.6 respectively) ; also that its 
achievement in rights was better than its achievement in attempts. 

EXERCISE 

Which of the two distributions given in Exercise 2, page 122, 
is the more variable and just how much? Precisely how do you reach 
your conclusion? For this, use the central tendencies and measures 
of deviation previously calculated. 

IV. CORRELATION 

Meaning of Correlation. Sometimes it is very advan- 
tageous to be able to show accurately and briefly the 
general relation between two distributions that have 
some common element or have been tested on the same 
thing. Thus it may be desired to get the relation between 
the two distributions obtained by testing the same group 
of cases on two different tests. For example, one may 
wish to know if a group of cities rank the same way on 
excellence of schools that they do on per capita school 
costs. 

Now it is evident in all such cases that the chances are 
very much against any situation where the cities would 
rank exactly the same or exactly opposite in both lists. 
Some will fall in exactly the same places, others exactly 
opposite, and still others will change indiscriminately. 
Consequently it is desirable to have some way of showing 
the extent to which the individual cases generally keep the 
same relative positions in the two distributions (that is, 
first in each, second in each, last in each, etc.), or, putting 
it another way, the extent to which the two distributions 
are correlated with each other. 

For a concrete example, let us take the following data 
on Cleveland ward schools, accumulated during the sur- 



174 



School Statistics and Publicity 



vey there. 1 The records of eighteen schools for the same 
grade on two qualities, which for our purposes we may call 
the A test and the E test, are given in Table 19. 

Table 19. Correlation Table Using Data from Eighteen 
Cleveland Schools 



School 



Brownell . . 

Clark . . . 

Marion . . . 

Detroit . . . 

Fullerton . , . 

Sackett . . . 
North Doan 

Bolton . . , 
East Boulevard 

Gilbert . . . 

Rosedale . . 

Landon . . . 

Lawn . . . 

Walton . . . 

Gordon . . 

Sibley . . . 

Waverly . . 

Halle . . . 



Record in 
A test 



32.8 
28.3 
28.0 
26.3 
26.1 
25.5 
25.0 
24.6 
24.0 
23.0 
22.9 
22.6 
21.9 
21.9 
21.7 
21.3 
20.8 
19.4 



Record in 
Etest 



10.0 
7.5 
6.8 
6.5 
7.5 
7.0 
7.0 
6.9 
5.9 
6.7 
8.5 
6.2 
5.3 
7.1 
6.6 
7.1 
5.2 



For our purposes it is not necessary to know which grades 
were used, what the tests mean, or how they were figured. 
With the records as given, the main question is : How did 
the rankings of the schools on the two tests correspond ? 
That is, did each school (or the majority of the schools) 

1 From some material turned over to the author to be used as 
practice work in his class on Statistical Methods Applied to Educa- 
tion at the University of Chicago, summer quarter of 1915. 



Measures of Relationships 



175 



Rating 
05 



so 



85 



20 



15 



fO 



Rating 
35 






30 



25 



20 



A Test 



IS 



E Te4+ 



z 
o 

a 
(D 



o Q 

<J5z 



d fc 



kj o 



-I (0 J 
CD lJ CD 



i! 



o — 



X 



Fig. 27. — Graph to Show Correlation Based on Data of Table 19. 



176 School Statistics and Publicity 

occupy the same, or approximately the same, relative 
position from the best on one test that it did on the 
other ? 

Graphic Methods of Showing Correlations. On look- 
ing at the figures we note that the poorest school on the 
A test is also poorest on the E test. But the best on the 
A test is only second on the E test. The second best on 
the A test is best on the E test. Similar observations 
might be continued without arriving at any idea of the 
relation existing between the distributions as wholes. 

But let us make two horizontal scales, one on each side 
of the page, running from to 35, and plot the two tables 
with the names of the schools on the base line, as in 
Figure 27. 

From this, it is apparent that in general the schools 
have about the same relative positions on the two tests, 
only there are some jogs in the lower curve showing that 
the relationship does not hold for every case. If it did 
hold absolutely for every case, the second curve would 
be parallel to the upper. 

Another way to study the relationship is to draw a 
scale for each test on opposite sides of the paper, prefer- 
ably making the two scales about the same height, and 
drawing a straight line across from the correct place 
on each scale to represent the performance of each school. 
This is merely an extension of the device used by Pro- 
fessor Haggerty.i If a school did as well on one test as 
on another, its line, of course, is parallel to the lines of 
most of the other schools. If it did not, the line will 
slant, depending upon which test it made the best record 
on. The Cleveland data are shown by this device in 
Figure 28. Observe that many of the lines are roughly 

iSeep. 172 



Measures of Relationships 



177 



parallel but that a fair number of them cut across each 
other and do not follow the general trend of relationship. 

Showing Correlation by Shifts of Cases within Parts 
of the Distribution. One very rough way to show 
correlation is to divide each . 
group by its median. Then 
look at each case to see if it ^^ 
is in the same half (upper or 
lower) in the second distri- 
bution as in the first. Call ^^ 
the upper half " plus " and 
the lower half " minus " in 
each distribution. Then get 
the percentage of like signs. 
This single figure which 
expresses the general rela- 
tionship between the two 
distributions is called the 
*' coefficient of correlation.' ' 

Graphically this relation- 
ship may be represented 15 
easily by writing the cases 
in one distribution in two ^ 

- „ 1 1 li? Fig. 28. — Graphic Representation 

colors, one tor each halt, say of Correlation between the Results 

black for the upper and red ^^ ^^^ a Test and those in the 

n ,1 1 rr\i 1 E Test for Eighteen Cleveland 

tor the lower. then keep schools. 

these same colors for each Each Une represents a school. (From 
• 1 1 T X •!_ data in Table 19, page 174.) 

case m the second distribu- 
tion, no matter where the case falls. Suppose for example 
that all cases in the upper half in the first distribution 
appear black and all in the lower half appear red. In 
the second distribution, if the cases fall exactly opposite, 
the upper half will be red and the lower black. If there 




178 



School Statistics and Publicity 



is a good deal of shifting, there will be some red and some 
black in each half. 

For the Cleveland data, this device is shown in Table 
20, the upper half of the first distribution being denoted 
by all capital letters, and the lower half by small letters, 
each case retaining its own type when it appears in the 
second distribution. 

Table 20. Like Signs Table for Correlation of Stand- 
ings OF 18 Cleveland Schools on Two Tests, Form I 



Median 



Rank and name of school 


Rank and name of school 




for A test 




for E test 


1 


BROWNELL 


1 


CLARK 


2 


CLARK 


2 


BROWNELL 


3 


MARION 


3 


London 


4 


DETROIT 


4.5 


MARION 


5 


FULLERTON 


4.5 


SACKETT 


6 


SACKETT 


6.5 


Gordon 


7 


NORTH DOAN 


6.5 


Waverly 


8 


BOLTON 


8.5 


NORTH DOAN 


9 


EAST BOULEVARD 


8.5 


BOLTON 


10 


Gilbert 


10 


EAST BOULEVARD 


11 


Rosedale 


11 


DETROIT 


12 


London 


12 


Rosedale 


13 


Lawn 


13 


Sibley 


14 


Walton 


14 


FULLERTON 


15 


Gordon 


15 


Lawn 


16 


Sibley 


16 


Gilbert 


17 


Waverly 


17 


Walton 


18 


Halle 


18 


Halle 



Twelve of the cases are in the corresponding half in the 
two distributions, so the coefficient of correlation is 
roughly i| or 66| per cent.^ 

1 This is not the real figure for this coefficient nor is the procedure 
used a strictly accurate one. But both are sufficiently accurate for 
many of the correlations that the superintendent will need until he 



Measures of Relationships 



179 



In using this method there must be as many ranks in 
each distribution as there are pairs of cases, — in this 
instance 18. If two schools tie, they get the average 
rank of the cases occupied by them. Note that Marion 
and Sackett on the E test are given 4.5 each because they 
tied for the 4th place and thus occupy the 4th and 5th 
ranks, or average 4.5 each. 

Another form of such a table keeps every case on the 
same line for the two distributions, as in Table 21. 



Table 21. Like Signs Table for Correlation of Standings 
OF 18 Cleveland Schools on Two Tests, Form II 



Median 



Rank and name of school 


Rank and name of school 




for A test 




for E test 


1 


BROWNELL 


2 


BROWNELL 


2 


CLARK 


1 


CLARK 


3 


MARION 


4.5 


MARION 


4 


DETROIT 


11 


Detroit 


5 


FULLERTON 


14 


Fullerton 


6 


SACKETT 


4.5 


SACKETT 


7 


NORTH DOAN 


8.5 


NORTH DOAN 


8 


BOLTON 


8.5 


BOLTON 


9 


EAST BOULEVARD 


10 
16 


East Boulevard 


10 


Gilbert 


Gilbert 


11 


Rosedale 


12 


Rosedale 


12 


London 


3 


LONDON 


13 


Lawn 


15 


Lawn 


14 


Walton 


17 


Walton 


15 


Gordon 


6.5 


GORDON 


16 


Sibley 


13 


Sibley 


17 


Waverly 


6.5 


WAVERLY 


18 


Halle 


18 


Halle 



By looking at the right hand or E test column, it is 
seen at once that there are three cases out of place in the 

has studied technical statistics much more thoroughly than is here 
attempted. 



180 School Statistics and Publicity 

upper part and three more in the lower. That is, 12 
cases out of 18 or 66f per cent of the cases are ahke 
in going into the same half of the two distributions. 

A much more elaborate device of the same nature, but 
covering two grouped distributions, was used by Pro- 
fessor Dearborn to show the relation between the achieve- 
ments of a group of students in high school mathematics 
and their achievements in college mathematics.^ He 
gave each student a number and placed this number in 
the proper column on a base scale carrying the grades 
from 60 to 100, so as to make a surface of frequency. In 
the high school mathematics distribution, all numbers 
below the first quartile were black ; all between the first 
quartile and the median, green ; all between the median 
and the third quartile, purple ; all above the third 
quartile, red. Thus there were four equal areas, each 
represented by a solid color (all numbers in it having the 
same color). The lowest fourth appeared as a black 
mass, the second fourth as a green one, the third 
fourth as a purple one, and the upper fourth as a 
red one. 

With the same number and color to represent the same 
student, the numbers were put in their proper places 
so as to make up the surface of frequency for achievement 
in university mathematics. This gave the second dis- 
tribution. All the students did not keep exactly the 
same positions that they had in the first distribution. 
Their switching around showed at once on the second 
distribution because the colors in one mass were not all 
the same but appeared variegated. The switching 
around in colors then gave a rough indication of the 

1 Dearborn, W. F. : " School and University Grades," in Bulletin 
of University of Wisconsin, No. 368, H. S. Series No. 9, p. 48 



Measures of Relationships 181 

relation or opposition (correlation) of the order of cases 
in the two distributions. 

Coefficient of Correlation. The foregoing ways of 
showing correlation are not exact enough and take too 
much space to show the desired relation with the best 
results in all cases. Consequently there have been 
devised various ways of showing this relation by the 
single figure called the coefficient of correlation. The 
calculation of this coefficient except by the very rough 
"like signs method " is generally very laborious, and the 
knowledge of what the results mean in the given circum- 
stances is very hard to acquire. 

The superintendent should not, in general, enter upon 
the calculation of this coefficient without much more 
extensive study of technical statistics than can be given 
here. Fortunately, there are few cases where the su- 
perintendent really needs to calculate this coefficient 
unless he is working on a thesis in some university, where 
he will be shown how to make the calculations. However, 
he does in his reading need to understand what the co- 
efficient of correlation means. Accordingly, we shall 
now explain clearly what it is, give some practical 
examples, and then merely append two ways of cal- 
culating it. 

Briefly, the coefficient of correlation is a number ex- 
pressing relationship between two distributions, based 
upon the changes in order or ranks of the different cases 
in the two groups. It ranges from + 1 (or + 100 per 
cent) expressing perfect agi^eement in order, through 
zero expressing only accidental or no relation, to - 1 (or 
- 100 per cent) expressing perfect opposition in order or 
ranking. In perfect relationship, the first case in distri- 
bution one will be the first case in distribution two ; the 



182 School Statistics and Publicity 

second case in distribution one will be the second case in 
distribution two, etc. In perfect opposition, the first case 
in distribution one is the last case in distribution two ; the 
second case in distribution one is the next to the last 
case in distribution two, etc. 

There is marked difference of opinion among statisticians 
and investigators as to the significance of the size of a 
coefficient of correlation. Any reader of educational articles 
in the last ten years which employ coefficients of correlation, 
will have noted the shifts and controversies on this point. 
Coefficients once thought large enough to be of importance, 
are now considered negligible. The size depends often 
upon the particular method of calculation. The importance 
of the size depends upon the particular method used for the 
particular distribution, a matter about which the authorities 
often differ. Probably the following from Professor Rugg 
is as safe as anything for the superintendent in his 
reading : 

This definition of limits (of high and low correlation) depends 
largely on the personal experience of the person making the interpre- 
tation. For example, it has been common for certain educational 
investigators to interpret arbitrarily a coefficient of .25 as an indi- 
cation of "high" positive correlation, and one of .40 as "very high." 
Others would interpret .25 as very low, and .50 as "marked" or "some- 
what high." Certainly, our educational conclusions must be colored 
by our arbitrary definition of such a coefficient. The experience of 
the present writer in examining many correlation tables has led him 
to regard correlation as "negligible" or "indifferent" when r is less 
than .15 to .20; as being "present but low" when r ranges from .15 
or .20 to .35 or .40; as being "markedly present" or "marked," 
when r ranges from .35 or .40 to .50 or .60 : as being "high" when it 
is above .60 or .70. With the present limitations on educational 
testing, few correlations in testing will run above .70, and it is safe 
to regard this as a very high coefficient.^ 

1 Rugg, H. 0. : Statistical Methods Applied to Education, p. 256 



Measures of Relationships 



183 



Examples of Coefficients of Correlation. The following 
examples of coefficients of correlation will be of signifi- 
cance to superintendents : 



Between what 
items 


Coefficient of 
correlation 


Where found 


Total cost per pupil 




Strayer, G. D. : City 


with 


+ .93 


School Expenditures, 


Teaching and supervision 




p. 95 


(49 city systems, 1902-3) 






Supervision 






with 


+ .15 


Same 


Eepairs (same) 






Payments for schools 




Elliott, E. C: Some 


with 


- .541 


Fiscal Aspects of Public 


Payments for interest 




Education, p. 85 


on city debt 






General merit of teachers 




Boyce, A. C.: "Methods 


with 


+ .54 


for Measuring Teachers' 


Neatness of room 




Efficiency," in 1 Uth Year- 
hook, National Society 
for Study of Education, 
Part II, p. 68 


General merit of teachers 






with 


+ .88 


Same 


General development of 






pupils 






General merit of teachers 






with 


+ .79 


Same 


Discipline 






Ability in reasoning 




Stone, C. W. : Arithmeti- 


with 


+ .73 


cal Abilities, p. 37 


Ability in fundamentals 






in arithmetic 






Ability in reasoning 






with 


+ .32 


Same 


Ability in addition 







184 School Statistics and Publicity 

Calculation of Coefficient of Correlation. The original 
treatment of correlation ended with the preceding exam- 
ples, but several readers of the manuscript recommended 
adding one or more simple calculations of a coefficient. 
For reasons which will not be clear to the casual reader, 
the coefficient of correlation calculated from the Cleveland 
data would not be reliable enough to be worth the effort to 
get it.^ These reasons center about the fact that the orig- 
inal measures were averaged from a number of individuals 
and not a simple measure for each school. But as it would 
take too much space and time to introduce and explain 
new sets of data, we shall assume that the data are suit- 
able for the purpose and proceed to give the calculations. 
The method of calculation is the only thing we care for here 
and it can be shown as well with the Cleveland data as with 
any other. The reader who has had no other train- 
ing in statistics is particularly cautioned against attempt- 
ing to calculate coefficients. The examples are given 
solely to clear up a reading knowledge of correlation. 

1. Spearman Rank-Order Method ^ 

6SD2 

P =1 ~~:; 77 

n(n2 - 1) 

S = sum of 

D2 = squares of differences in ranks (must be same number of ranks 

in both distributions) 

n = number of pairs in distributions 

P = coefficient of correlation 

1 See J. F. Kelley : Teachers' Marks, p. 88 

2 The formula here given is only one factor in the correct formula, 
but it is sufficiently accurate for administrative problems. See Thorn- 
dike : Mental and Social Measurements, p. 167 



Measures of Relationships 



185 



From page 178 we get D^ as follows : 





D 


D' 


Brownell ; 


1 


1.00 


Clark 


1 


1.00 


Marion 


1.5 


2.25 


Detroit 


7 


49.00 


Fullerton 


9 


81.00 


Sackett 


1.5 


2.25 


North Doan 


1.5 


2.25 


Bolton 


.5 


.25 


East Boulevard 


1 


1.00 


Gilbert 


6 


36.00 


Rosedale 


1 


1.00 


London 


9 


81.00 


Lawn 


2 


4.00 


Walton 


3 


9.00 


Gordon 


8.5 


72.25 




3 


9.00 


Waverly. 


10.5 


110.25 


Halle. 





0.00 






462.50 



p = l- 



6 X 462.50 



= +.52 



18(324 - 1) 

2. The Pearson Method 

The Pearson coefficient of correlation is calculated by the following 
formula : 

Coefficient of Correlation or r= (^ • V) — 

Vsx2 VS2/2 



S = algebraic sum 
X = deviations in one distribution 
y = deviations in other distribution 
X ' y = product of deviations for corresponding case. 



186 



School Statistics and Publicity 



The calculation on the Cleveland data is as follows, the deviations 
being taken from the median : 



School 


Score 


X 


x' 


Score 


y 


y' 


-\-xy 


—xy 


Brownell . 


32.8 


+9.3 


86.49 


8.8 


+ 1.85 


3.42 


+ 17.21 




Clark . . 


28.3 


+4.8 


23.04 


10.0 


+3.05 


9.30 


+ 14.64 




Marion 


28.0 


+4.5 


20.25 


7.5 


+ .55 


.30 


+ 2.48 




Detroit 


26.3 


+2.8 


7.84 


6.8 


- .15 


.02 




- .42 


FuUerton . 


26.1 


+2.6 


6.76 


6.5 


- .45 


.20 




-1.17 


Sackett 


25.5 


+2.0 


4.00 


7.5 


+ .55 


.30 


+ 1.10 




N. Doan . 


25.0 


+1.5 


2.25 


7.0 


+ .05 


.003 


+ .80 




Bolton . . 


24.6 


+ 1.1 


1.21 


7.0 


+ .05 


.003 


+ .06 




E. Blvd. . 


24.0 


+ .5 


.25 


6.9 


- .05 


.003 




.03 


GUbert . 


23.0 


- .5 


.25 


5.9 


-1.05 


1.10 


+ .53 




Rosedale . 


22.9 


- .6 


.36 


6.7 


- .25 


.06 


+ .15 




London . 


22.6 


- .9 


.81 


8.5 


+ 1.55 


2.40 




1.40 


Lawn . . 


21.9 


-1.6 


2.56 


6.2 


- .75 


.56 


+ 1.20 




Walton . 


21.9 


-1.6 


2.56 


5.3 


-1.65 


2.72 


+ 2.64 




Gordon 


21.7 


-1.8 


3.24 


7.1 


+ .15 


.02 




- .27 


Sibley . . 


21.3 


-2.2 


4.84 


6.6 


- .35 


.12 


+ .77 




Waverly . 


20.8 


-2.7 


7.29 


7.1 


+ .15 


.02 




- .41 


Halle . . 


19.4 


-4.1 


16.81 


5.2 


-1.75 


3.06 


+ 7.18 






Med. 23.50 


190.81 


Med. 6.95 


23.61 


+48.04 


-3.70 



I 



S xy 



44.34 



V 



2;x-^ 



Vs 



= + .66 Ans. 



V190.81 V23.61 

(13.81) (4.86) 

The Pearson coefficient in actual work is often figured from the 
guessed average, the result being corrected by a formula given in this 
Thorndike and Rugg references. This saves time because the devia- 
tions will thus always be integers or steps denoted by integers, and the 
handling of these will be much easier than when deviations with deci- 
mals are used as in the example just given. 

REFERENCES FOR SUPPLEMENTARY READING 

King, W. I. Elements of Statistical Method, Chapters XIV, XVI, 

XVII, XVIII. 
Rugg, H. O. Statistical Methods Applied to Education, Chapters 

VII, VIII, IX. 
Thorndike, E. L. Mental and Social Measurements, Chapters X, XI. 



CHAPTER VIII 
SUPPLEMENT ON STATISTICAL TREATMENT 

The preceding chapters cover most of the problems 
which the superintendent will encounter in working up 
his school statistics, until he reaches the actual presenta- 
tion to the public. But there are three other practical 
problems, not directly connected with any of the fore- 
going problems nor with one another, which will be of 
interest to him. These are : 

1. How may one insure reliability in the statistical results or at 
least know about how accurate they are? 

2. What special economies may be employed to reduce the in- 
evitably large labor involved in statistical calculations? 

3. How may data given in ranks only (for example, the decisions 
of judges in a contest) be easily combined? 

These problems will now be discussed in order. 

I. RELIABILITY OF STATISTICAL RESULTS 

The aim throughout this book has, of course, been to 
produce reliable results, and cautions for this purpose 
have been included in many places. But in spite of 
these, the reader will probably wish to know how to 
avoid the numerous errors in adding, multiplying, omit- 
ting figures, etc., that all of us understand are liable to 
creep into any numerical work. 

Variable Errors. The simplest way to regard these is 
to distinguish between variable and constant errors. A 

187 



188 School Statistics and Publicity 

variable error is one that is as likely to occur one way as 
the other and so the various results will offset each other. 
For example, in adding, the average person is as apt to get 
results too large about as often as he gets them too small 
and in about the same amounts. In any problem in- 
volving many additions, these errors will balance each 
other and so the result may be assumed to be fairly correct. 
The principle is exactly that involved in the common 
practice of discarding the last decimal place by adding 1 
to the preceding place if this decimal is 5 or more, and 
throwing it away altogether if it is less than 5. Or it is 
the same principle as that behind the old business prac- 
tice of settling a bill to the nearest five cents, — paying 
$1.25 if the bill comes to $1.23 or only $1.20 if it comes to 
$1.22. In both of these, it is assumed that in the long 
run things will even up and no essential injustice or in- 
accuracy will result. 

Most of the errors in ordinary statistical calculations 
are variable ones and, if reasonable care is given the work, 
they may then be ignored. But they may be still further 
removed by the devices of utilizing students or assistants 
to check each other's work, or of going through each pro- 
cess at least twice. If the result comes out the same each 
time, it may be safely assumed to be correct. If, after 
repeated trials, the results all vary slightly, the average 
of them is safe enough for practical purposes. 

Constant Errors. A constant error, on the other hand, 
is one that always tends in the same direction. The 
marks in deportment given healthy boys by a well- 
poised, healthy teacher would in the long run be sub- 
stantially correct. If he were ill one month and cut 
them very low, the result would probably be offset by 
correspondingly high marks in some month when he was 



Supplement on Statistical Treatment 189 

feeling well. That is, his variable errors would offset 
each other. But if these same boys were under a nervous 
or irritable teacher, his errors would probably all tend 
in the direction of giving them very low marks. That is, 
he would have a constant error. Similar constant errors 
may be expected in such cases as these : reports on 
number of tardies, because teachers easily forget to mark 
tardies but are very unlikely to mark present pupils as 
absent at the time of noting tardies; reports on training 
of teachers, because teachers are apt to try to make their 
training seem as extensive as possible and very unlikely 
to understate it ; itemized and verified expense accounts, 
which usually tend to be less than the actual expenditure 
because of the general tendency to forget to put down 
items, and the absence of intent to be dishonest. 

There is no general rule for avoiding constant errors, 
except possibly to watch for fallacies in sampling. Con- 
stant errors are never , negligible but " skill in avoiding 
them is due to capacity and watchfulness far more than 
to knowledge of any formal rules." ^ 

Weighting Results. Sometimes the attempt is made 
to avoid errors by weighting certain factors to give them 
more influence in determining the results. This should 
never be attempted in the belief that it will give more 
accurate results, except by one especially trained in 
statistics. " Bowley gives a rule that is satisfactory 
for most cases that occur in practice; namely, to give 
your attention to eliminating constant errors and not to 
manipulating weights." ^ 

Estimating Reliability of Results. All complete treat- 
ments of statistical methods include formulas for estimat- 

1 Thorndike, E. L. : Mental and Social Measurements, p. 209 

2 Ibid., p. 212 



I 



190 School Statistics and Publicity 

ing the reliability of the various averages, measures of 
dispersion, or coefficient of correlation. But in practi- 
cal school statistics there is no need for these. The 
superintendent had far better employ his time in seeing 
that he avoids the errors in collecting and manipulating 
his data that common sense and reasonable care would 
indicate. Beyond this, he needs to know only that, if I 
constant errors are avoided, accuracy is increased by 
taking more measures of an item, or samples from a 
distribution. From a random sampling of any material 
number of cases, accuracy increases as the square root 
of the number of samples. That is, 400 samples would | 
double the accuracy obtained by taking only 100 samples. 

II. SPECIAL ECONOMIES IN CALCULATION 

Many economies have been indicated in connection 
with the particular processes to which they pertain. 
But there are several others that merit special notice. 

Use of Cross-Section Paper. Plain cross-section paper 
is very valuable for saving time in getting figures lined 
up in columns and in keeping horizontal lines straight. 
It also lessens eyestrain. The kind with a little square 
just large enough for one digit is best. This paper is 
especially valuable for saving copying, because a part 
of a calculation that is correct or that needs to be shifted 
can easily be cut out, removed to where it is needed, and 
quickly pasted into line there. 

Checking in from Both Ends. The old device of 
checking addition by adding the column up and then down 
until the results agree is very helpful for insuring accuracy. 
The device of calculating the median by going up, and 
then by coming down, until the two results agree, is 
equally serviceable. 



Supplement on Statistical Treatment 191 

Use of Steps in Distributions Involving Large Numbers. 
If the distribution involves large numbers, time can 
usually be saved by employing steps instead of the actual 
numbers for getting the average from the guessed average, 
for getting the deviation measures, etc. Thus, if one is 
studying superintendents' salaries grouped by hundreds, 
and the guessed average is 1500, the deviations do not 
have to be given in hundreds. The 1600 group could be 
counted as one step above ; the 1400 group, one step 
below, etc. The deviation measure finally calculated would 
then be in steps ; if multiplied by $100, the value of a step, 
the real deviation measure would be at once secured. 
This method obviously saves much labor in copying zeros. 
It would save much more labor in multiplication, if the 
value of a step was some such number as 25, 5.5, etc. 

One Step All the Way Through. It is much easier men- 
tally and insures accuracy as well, to perform the same 
process all the way through at one time. Thus, all data 
may be copied to the end ; all multiplications may be 
made at one step ; all additions made at another ; all 
squares found at another, etc. 

Calculating Devices. Various tables may be utilized 
to save time in multiplying, dividing, squaring, or ex- 
tracting roots. These are mentioned in the bibliography. 
The writer has used Crelle for years, but this is not now 
obtainable in this country. The short tables given in 
Thorndike's Mental and Social Measurements can often 
be utilized by employing the step method as given 
above. These tables are not at all difficult for even school 
children to use. Some years ago, the twelve-year-old 
son of a friend of the writer regularly used the Thorndike 
tables in computing the batting averages of his favorite 
baseball stars. 



I 



192 



School Statistics and Publicity 



Most superintendents will be able to get the use of an 
adding machine at the city hall, the county courthouse, 
or some bank, if the school system does not have one. 

Utilizing Students in Calculations. What was said 
about the value to students of collecting data (see page 
86), applies even more to practice on the calculations 
involved in working them up. Such data afford the 
finest sort of laboratory material for the upper arithmetic 
and high school mathematics classes. And much of the 
work is very valuable for clerical practice. 

m. COMBINING DATA GIVEN IN RANK ORDER ONLY 

The school administrator often faces a problem which 
involves a summary for the relative merits of a number 
of items that have been graded on different phases with 
widely varying standards, or by different judges with 
varying standards. Thus, he may wish to find the vale- 
dictorian in the high school from the grades given by 
many teachers. He may wish to find the best contestant 
in a debating contest from decisions given by several 
judges. Or he may wish to find which ward school is 
on the whole the best, as judged by results from standard 
tests in arithmetic, composition, spelling, and handwrit- 
ing, all given on very different scales. In all of these 
he must find the relative standing or rank of several 
people or schools. How can he combine the rankings 
given from one judge or one test with those from another 
judge or another test ? 

These problems may be conveniently classified for 
practical purposes as being of five types : 

1. How may rankings from different distributions, having the same 
number of ranks and being considered as of equal value, be combined ? 
For example, the decisions of judges in a contest. 



I 



Supplement on Statistical Treatment 193 

2. How are tie rankings within one or more distributions to be 
treated when being combined with rankings from the other distribu- 
tion? 

For example, one judge in the contest gives a tie vote on two 
contestants. 

3. How are rankings from one distribution to be combined with 
rankings from other distributions which have a different number of 
ranks ? 

For example, a rank from a class of 12 in chemistry is to be 
combined with a rank from a class of 20 in English ; or rankings 
on daily grades from a class of 20 with rankings for only 18 
who took the examination. 

4. How are rankings from different distributions to be combined 
when it is desired to give special weight to one or more of the dis- 
tributions ? 

For example, it is desired to count rankings on daily grades 
three times as much as rankings on examination, to secure the 
rankings for final grades. 

5. How is absolute unfairness on the combinations to be avoided? 

For example, in the author's class in statistical methods at one 
time, tests were being given on which the class were ranked. 
The rankings were obtained from two separate distributions, 
one of rankings on speed, the other of rankings on accuracy. 
Speed and accuracy were counted equal. On one test, a stu- 
dent at the beginning handed in a blank paper, claiming first 
rank on speed and being willing to take lowest rank on accuracy. 
With thirty in the class, he counted on a final or average ranking 

of — X_ , or 15.5, or about the middle of the class. 
2 

These will be discussed in order. 



1. Distributions Having Same Number of Ranks 

This is managed by adding the various ranks ; then 
ranking the items in order of the sums, giving the one 
with the smallest sum a final rank of 1, the one with the 
next smallest sum a rank of 2, etc. 



194 



School Statistics and Publicity 



For example, take the illustration of the judges in a contest, given 
on page 12, changing some of the figures to avoid any ties. This 
would keep the number of ranks the same for each judge. The orig- 
inal marks would be : 




Changing these to ranks we would have : 



Contestant 


Ranked by 


Ranked by 


Ranked by 


Sum of 


Final 


Judge A 


Judge B 


Judge C 


ranks 


rank 


1 


10 


6 


10 


. 26 


9 


2 


1 


8 


5 


14 


5 


3 


8 


5 


7 


20 


7 


4 


2 


1 


6 


9 


lor 21 


5 


7 


7 


8 


22 


8 


6 


3 


3 


4 


10 


3 


7 


4 


2 


3 


9 


lor2i 


8 


6 


4 


2 


12 


4 


9 


5 


10 


1 


16 


6 


10 


9 


9 


9 


27 


10 



If the judges had rated the contestants by rankings in the first 
place, the result could, of course, have been obtained much more 
quickly. 

1 These tie, and average 1.5 each. See page 195. 



Supplement on Statistical Treatment 195 

2. Tie Rankings 

This is handled by changing the tie rankings so as to 
make the same number of ranks in each distribution, 
then proceeding as in 1. 

For example, the rankings from the marks of the judges as given 
on page 12 would be : 





Ranked by 


Ranked by 


Ranked by 




Judge A 


Judge B 


Judge C 


1 


8 


4 


5 


2 


1 


5 


2 


3 


6 


4 


3 


4 


2 


1 


2 


5 


6 


4 


4 • 


6 


3 


2 


1 


7 


4 


2 


1 


8 


5 


3 


1 


9 


5 


5 


1 


10 


7 


4 


3 



Obviously, these could not be combined as they now are, for two 
judges used only five ranks and the other one had eight, so that rank 
1 does not mean the same in the three distributions. 

The trouble will be removed by changing the rankings so as to 
make each judge cover the entire ten ranks. Thus, in the original 
table on page 12, Judge A has given Contestants 8 and 9 each the 
rank of 5. This means he has really made them take up ranks 5 and 
6, or average 5.5 in rank. This leaves Contestants 3 and 5 as ranked 
by him to occupy ranks 7 and 8 or average rank 7.5 each. Proceeding 
thus, the real rankings from the figures on page 12 would be those 
given on the following page. 

It would, of course, have been easier to require the judges to give 
no tie rankings, but if they should insist on turning in tie ranks, the 
only safe procedure is that given. 



196 



School Statistics and Publicity 





Real rank 


Real rank 


Real rank 


Sum 


Final 
rank 


Contestant 


by 


by 


by 


of 




Judge A 


Judge B 


Judge C 


ranks 


1 


10 


6.5 


10 


26.5 


10 


2 


1 


9.5 


5.5 


16 


5 


3 


7.5 


6.5 


7.5 


21.5 


7 


4 


2 


1 


5.5 


8.5 


2 


5 


7.5 


6.5 


9 


23 


8.5 


6 


3 


2.5 


2.5 


8 


1 


7 


4 


2.5 


2.5 


9 


3 


8 


5.5 


4 


2.5 


12 


4 


9 


5.5 


9.5 


2.5 


17.5 


6 


10 


9 


6.5 


7.5 


23 


8.5 



3. Different Number of Ranks in the Distributions 

This is practically the same as the problem of a tie 
vote, except that here, instead of taking the average of a 
set of items and getting the average rank for each, one may 
have to take the average range of an item to get its rank. 

For example, suppose that one wishes to combine 
ranks of a class of 12 in chemistry with those from a class 
of 20 in English. The ranks in chemistry should cover 
20. This will make one rank in chemistry cover 1.6 
ranks in English. If the second and third pupils in 
chemistry tied, they would cover ranks 3.2 and 4.8, or 
average rank 4 each. 

In the case where 20 pupils were ranked on daily grades, 
but only 18 took the examination, a similar procedure 
could be followed. But a much shorter approximate 
result might be obtained by making the 18 pupils get 
ranks 1 to 18 on examination, giving 19.5 each on examina- 
tion to the two absent pupils. The question of passing 
pupils absent from examinations would have to be 
decided entirely apart from the matter of ranks. 



Supplement on Statistical Treatment 197 

4. Weighting Factors Given in Ranks from Different 

Distributions 

The way to do this is to arrange the factors by ranks, 
with the same number of ranks in each distribution. Then 
multiply the factor to be weighted by the number of 
times it is desired to weight it. The ranks can then be 
added as in previous instances. 

For instance, suppose pupils have been rated as follows on exam- 
ination and daily grades : 



Pupil 


Ranks on 


Rank on 




daily grade 


examination 




A 


1 


5 




B 


2 


2 




C 


3 


4 




D 


4 


3 




■ E 


5 


1 





If it is desired to count the daily grades three parts and the examina- 
tion grade one part in determining the final rating, the original ratings 
for daily grades should be multiplied by 3 ; thus : 





Weighted 


Rank in 


Sum of 
ranks 


Final 


Pupil 


daily grade 


examination 


rank 




rank 


grade 






A 


3 


5 


8 


1.5 


B 


6 


2 


8 


1.5 


C 


9 


4 


13 


3 


D 


12 


3 


15 


4 


E 


■ 15 


1 


16 


5 



5. Avoiding Absurdities in Combining Ranks 

No mechanical device or general advice will take the 
place of common sense in avoiding absurdities such as 



198 School Statistics and Publicity 

that of the student mentioned on page 193, who handed 
in a blank paper. In this particular instance, the com- 
mittee of students appointed to consider the case rightly 
decided that common sense and fairness demanded that 
the student was not entitled to any consideration what- 
ever until all those who had really tried to do something 
had been assigned ranks. Similarly, the matter of giving 
any ranks whatever to the two students who failed to 
take the examination in the instance given on page 196 
would have to be decided entirely apart from the mechan- 
ical problem of adjusting the ranks. The whole matter 
of ranking might be left until these two had taken the 
examination. But as such a delay on reporting grades 
or ranks is hardly practicable in most places, it would be 
simpler to rank all the others and later, after the two had 
taken the examination, to give them tie ranks that would 
place them approximately where they belonged. Much 
energy is sometimes wasted in trying to combine rankings 
where the same subjects did go wholly through two differ- 
ent sets. Thus a student was worried over what to do 
with an experiment with fourteen white rats where one died 
in training so that he had to train a new fourteenth rat. 
A junior high school principal studying twenty pupils had 
only nineteen of these in both distributions and wished to 
take a new twentieth one for the second test. The simplest 
and safest procedure is to take only the cases that appear 
all the way through. 

EXERCISE 

Find the relative standing of each of the twenty cities for rights 
in arithmetic, on the four processes combined, from the data on the 
following page. 



Supplement on Statistical Treatment 199 

Rankings of Twenty Indiana Cities on Rights in Arithmetic 
FOR THE Courtis Tests in Fifth Grade Only ^ 







Ranks 




City 


















Addition 


Subtraction 


Multiplication 


Division 


1 


8 


6.5 


13 


14.5 


2 


19.5 


17.5 


20 


20 


3 


3 


4 


3 


6 


4 


12.5 


16 


15 


7.5 


5 


19.5 


19 


19 


19 


6 


5.5 


5 


9 


14.5 


7 


16.5 


12.5 


10.5 


9 


8 


16.5 


14.5 


7.5 


3 


9 


15 


20 


18 


17.5 


10 


7 


3 


4.5 


13 


11 


5.5 


9.5 


13 


16 


12 


14 


9.5 


6 


11 


13 


2 


1.5 


1 


1 


14 


12.5 


9.5 


10.5 


10 


15 


9.5 


14.5 


16.5 


12 


16 


18 


17.5 


16.5 


17.5 


17 


1 


1.5 


2 


2 


18 


9.5 


-9.5 


7.5 


4.5 


19 


11 


6.5 


13 


7.5 


20 


4 


12.5 


4.5 


4.5 



REFERENCES FOR SUPPLEMENTARY READING 

Rugg, H. O. Statistical Methods Applied to Education, pp. 134-147. 
Thorndike, E. L. Mental and Social Measurements, pp. 51-59, 
Chapters XII, XIV. 

1 Adapted from M. E. Haggerty's "Arithmetic: A Cooperative 
Study in Educational Measurements," Indiana University Bulletin, 
Vol. XII, No. 18, p. 443. 



CHAPTER IX 

USELESSNESS OF STATISTICS IN CURRENT 
SCHOOL REPORTS 

I. THE SITUATION 

Thoughtful school men will agree with Professor 
Hanus in his statement that many of the ordinary presen- 
tations of statistical material on schools to the public are 
useless. 1 In his investigation, which included a random 
selection covering the entire country, he found that the 
reports studied by him were '' vague in purpose, miscel- 
laneous in subject matter, and hence ineffective/' ^ jjg 
found that many of the tables were printed just as they 
had been tabulated by the superintendent; that exactly 
50 per cent of the reports studied had no adequate inter- 
pretations of tables; that only 42 per cent contained 
any comparative statistics whatever. 

1 Compare, for example, the statement of Superintendent Giles 
of Richmond, Ind. : "Much of the labor expended on school reports 
is lost because they are not in a form useful for comparison." This 
is contained in an article in Educational Administration and Super- 
vision, Vol. II, p. 305, where he attempts "to make available for ad- 
ministrative purposes a part of the annual reports of city superin- 
tendents in Indiana to the State Department of Public Instruc- 
tion." 

2 Hanus, Paul H: "Town and City Reports" (more particularly 
superintendents' reports), School and Society, Vol. Ill, p. 196 

200 



statistics in Current School Reports 201 

n. CAUSES OF THE USELESSNESS 

Hanus makes the point that '' the useless tables of 
statistics are often quite useless because they are mere 
collections of working data unrelated and uninterpreted, 
and also because they pertain only to the year under re- 
view." 1 Or as he puts it in another place, '' without inter- 
pretation or discussion, many statistics are meaningless 
to most readers, whether lay or professional." ^ 

Hanus believes that a part of this confusion exists 
because one report is gotten out for several classes of 
people who really need separate reports. He says that 
the superintendents must give information concerning 
schools to three classes : (1) the school board ; (2) the 
teaching staff; (3) the public. The so-called annual 
report is usually gotten up for the three classes with one 
or two more or less consciously in mind, but it often 
fails to be serviceable to any of the three. '' Reports to 
the board or staff should always contain both the statistical 
summaries and the details from which they are derived. 
. . . Reports to the people should contain only sum- 
maries, by items, of course, and for successive years or 
periods; and special pains should be taken to interpret 
them so clearly that he who runs may read." ^ The 
report to the people should be ''an abstract or digest of 
the reports to the board, brief but comprehensive and 
clear." In addition to the three classes noted by Hanus 
in his thoroughly representative reports, some super- 
intendents apparently write for two other classes : 
(1) the state or national educational officials, (2) students 
of education, as indicated by Snedden and Allen. * But 

1 Ibid., p. 196 2 j})ifi^^ p 195 3 /^^^^.^ p, 195 

^ School Reports and School Efficiency, pp. 4, 5 



202 



School Statistics and Publicity 



this, of course, only makes the reports all the worse for 
the public. 

Whatever the relative values of the different reports, 
the superintendent should consider very carefully the 
preparation of the school statistics for the public. Ifl 
this is to be done effectively, he must first master the 
statistical material in hand and get hold of the significant ■ 
points to be found in the data. For help on this, hei 
may consult previous chapters. But his time spent in 
preparing the special report for the public will not be| 
lost for either himself or his teachers. Honest statistics 
gotten up to influence the public will influence school 
men and teachers all the more strongly because the 
striking points set forth so vividly in the popular presen- 
tation will make more impression on teachers. The old 
saying that we never completely know a thing until we 
can tell it effectively to some one else applies here. The 
superintendent never fully understands his own statistics 
and their relations until he can present the significant 
points effectively to the public. In short, to paraphrase 
Snedden and Allen, in the main, the methods that will 
give the maximum of publicity on school statistics will 
probably result also in providing the most effective statis- 
tical basis for school administration. ^ 



I 



III. DEVICES FOR EFFECTIVE PRESENTATION 

But granted that the superintendent realizes the in- 
effectiveness of his school statistics and is determined 
to issue a good report specifically for the public, what 
devices are available for him? There are only three 
main devices for this purpose. He may graph his statis- 
^ School Reports and School Efficiency, p. 8 



Statistics in Current School Reports 203 

tics; he may translate them into words; or he may 
tabulate them. Each device is best suited for certain 
conditions. Each has its strengths and weaknesses. 



i 1. Graphic Presentations 

A graph or drawing can be so constructed as to show 
at a glance the significant things in a mass of technical 
data. Then, too, it is so very simple that, if properly 
made up, it can be readily grasped by the average man 
who is totally unversed in the intricacies of statistical 
manipulation. In general, however, it cannot be 
accurately remembered or reproduced without consider- 
able effort. 

2. Translation 

By translation is meant the transferring or inter- 
pretation into ordinary language of statistical ideas, 
relationships, and results. These need to be translated 
for the ordinary man before they can be clear to him, 
much less affect him to any appreciable extent. Such 
translation is difficult because sometimes it is as hard to 
give statistical results through mere words as it is to 
describe a painting in words only. But the superintendent 
in employing translation has only the same problem as 
has any able editorial writer, any successful campaign 
orator, any writer on agricultural or food topics, or any 
writer for a society engaged in propaganda on a large 
scale. Some of the best effects produced by such persons 
are managed through translating statistical material in 
word language only. A good translation can be easily 
remembered and used in a speech or article without any 
special preparation. 



204 School Statistics and Publicity 

3. Tabulation 

Probably on the whole the least serviceable device is 
tabulation. The ordinary tabulations and masses of 
statistics in most school reports are of very little value 
for the public. The significant facts do not stand out in 
tables of this sort. It requires much labor for any 
person other than the maker of an ineffective table to 
get anything out of it. Certainly it would be unreasonable 
to expect the average man to waste any time on tables 
of this nature, even if he can be persuaded to read the 
rest of the report. In such tables, comparisons with 
similar data are usually impossible. If facts about a 
number of ward schools in the system are given, the 
arrangement may be alphabetical, and the reader cannot 
find whether School A is better than School B without 
some labor. Norms or standards with which any item 
in the table might be compared may be left out. There 
may be no units of measurement. Often great masses 
of facts related slightly or not at all to each other are 
thrown together in the same table. 

Even if helpful devices are employed in tabulation, 
or if the table is arranged like a scale with the largest 
magnitude at the top and constantly decreasing, or if 
heavy type is used for the home city or special school, or 
if the table is simplified by breaking it up into parts and 
so on, still the graphs and translation are probably best 
for the public' In either of the latter, the significant 
facts are outstanding and may be grasped by the reader 
with less labor than he will have to use in understanding 
even good tabulations. The impression that the graphs 
and translation make is more striking and lasting. 

However, tabulations are necessary in the statistical 



Statistics in Current School Reports 205 

process because they must precede the graph or trans- 
lation. The graph is only a pictorial representation of 
the significant facts in a tabulation. The translation is 
only a word representation of the same thing. Often- 
times we do not know what the things we wish to graph 
or translate for the public are, until we have made the 
proper tabulation. Consequently we shall take up in 
detail the matter of tabulating school statistics, following 
it with treatments of graphs and translations. 



CHAPTER X 

PRESENTING TABULATED STATISTICS TO THE 

PUBLIC 

I. POSSIBILITIES OF USING TABULATIONS OF STATISTICS 
TO INFLUENCE THE PUBLIC 

As previously shown, one of the chief reasons for the 
many useless tables of statistics in school reports is that 
the average school superintendent tends to publish 
without modification the tables he gets up for his own 
use or for state and national requirements. ^ But the 
mere publishing of such tables is about the worst thing 
he can do if he is trying to influence his public. It is 
practically certain that the majority of people who read 
school reports at all tend to skip conscientiously most of 
the tabulated matter of this nature. 2 

What, then, is the object of preparing a tabulation for 

1 See p. 201 

2 This does not in any way militate against the contentions of 
Professor Thorndike and others that educational investigators should 
get into the habit of publishing in full the data secured in their studies. 
Such complete publications are highly desirable, but not for the 
general public. They should appear in educational periodicals, mono- 
graphs, dissertations, and similar places. If included in reports to 
the public, they should at least be in appendices where they will not 
prevent the citizen from reading the parts of the report that will in- 
fluence him. But even thus they will probably weaken the effect 
of the other parts upon him. 

206 



Presenting Statistics to the Public 207 

the public? The original intention undoubtedly is to 
give the public quickly and easily a bird's-eye view of a 
great mass of data which are entirely unintelligible 
until collected and systematized. Again, tabulation 
achieves a great economy of space through combining 
many items which had been isolated up to that time. 
This saving of space will not be achieved without some 
care. For example, it is possible to make numerous 
separate tables with all the headings repeated for each, 
as in reporting facts about all the grades, so that the 
total occupies more pages than straight reading matter 
giving the same facts would take up. But in general, the 
argument is that, if the ordinary reader can understand 
the tabulation and will read it, he is more likely to be 
influenced by it than by the pages of print necessary to 
give the same idea. There is, however, a great danger in 
undue condensation, especially in getting too many 
items into one table. For example, the 1914-15 report 
for Des Moines has one table with ninety different 
columns.^ The ordinary reader can hold in mind only 
a few different items at one time. It is, therefore, out 
of the question to count on such a complicated table's 
influencing the public. And it is probably unwise to 
insert a table of this kind in a report intended even 
partially for public consumption. A third object in 
tabulation is the aid to logical thinking and presentation 
which it affords for all work capable of expression in 
numbers. Most of us recall how our science teachers 
insisted that the extent to which we mastered scientific 
thinking would be shown by our laboratory reports. 
Probably these reports first acquainted us with tabulation. 

^Annual Report of the Des Moines Public Schools, for the year 
ending July 1, 1915, opposite p. 82 



208 



School Statistics and Publicity 



But the distinction between tabulations intelligible to 
school men or experts and those easily understood by the 
public cannot be overemphasized. Tabulation, although 
a very useful device for work, does not come from 
instinct. It must be acquired, a truth often forgotten 
by the expert tabulator. Most of us probably recall 
some good mathematician who makes a very poor in- 
structor because he has forgotten how painfully slow was 
his own mastery of some of the work he is teaching. He 
has consequently no patience with his students. Similarly 
a superintendent familiar with tabulation is very apt to 
forget that the public is not so familiar. It is true that 
many a business man makes much use of tabulation in 
his work. He has been trained to do this, however, for 
no man can successfully conduct a business involving 
many details unless he 'knows how to collect and con- 
solidate data concerning his complex enterprise, in what 
corresponds to tabular form. 

But even an expert, trained to see data thrown into 
certain conventional or familiar forms, is easily upset 
if confronted with a tabulation differing from the form 
to which he is accustomed. If a table is arranged with 
magnitudes running from low to high instead of from 
high to low, or if the order of years runs from the most 
recent to the most remote instead of vice versa, or if 
any similar change is made, the expert accustomed to 
his own different tabulations will often hesitate and 
become lost for a while. It may take him some time to 
understand the new forms. The ordinary man would 
not give even that much time to trying to understand 
them. He would pass by the table the instant that he 
could not easily grasp its meaning. 

It should be evident by now that the making of tab- 



Presenting Statistics to the Public 209 

ulations to influence the public is no easy task. It is 
indeed of so difficult a nature as to require study. We 
shall now proceed to make this study, taking up first the 
problem of how to give a bird's-eye view through tab- 
ulation, and following it with a consideration of how to 
get up a series of tables. The reader should clearly 
understand that all this presupposes an acquaintance 
with the preliminary treatment of blanks and tabulating, 
pages 71-81. 

II. HOW TO GIVE A BIRD'S-EYE VIEW THROUGH 

TABULATION 

Many points are to be considered in making a table for 
the public. But it should before all else strive to give 
the reader quickly a bird's-eye view of the data. Among 
the things required for this are : 

1. A Very Carefully Worded and Specific Heading 

The heading should be so well chosen as to give the 
key to the understanding of the table, even if no other 
explanatory material follows. 

As examples of incomplete headings, the following from the South 
Bend Survey are pertinent: 

(a) Table headed "Value of School Property" (page 13). 

It contains the dates of the erections of the various build- 
ings, sizes of the lots, value of buildings and equipment. 

(6) Table headed "Registration" (page 15). 

It contains not only the registration by years but the in- 
crease in registration and average daily attendance. 

(c) Table headed "Cost of Fuel, 1912-13" (page 21). 

This is by ward schools and contains the number of tons 
each school used, the total cost in a lump, and the cost per 
pupil. 

In none of these does the title really specify what is in the table. 



210 School Statistics and Publicity 

2. Comparatively Few Classes of Facts in One Table 

Of course, it is hard to say just how many classes of 
facts should go into one table. But it is safe to say that a 
table containing more than four such classes will not be 
read by the average man. By far the greater portion of 
the tables in the best school surveys are concerned with 
only one class of facts, and those including as many as 
three classes are very rare. No one will question the 
statement that the best methods of presenting school 
statistics to the public have so far appeared in these 
surveys or in annual reports that are virtually surveys. 

3. Careful Grouping of the Classes That are Used 

A complete running list of items is of little value. It 
corresponds to the term of derision used toward a person 
who puts too many items into a story or explanation, 
'' total recall." 

Thus in the Birmingham, Alabama, report for 1914-15 there is a 
bird's-eye tabulation of five city school systems, covering twelve 
items (pages 18 and 19). In the Des Moines report for 1914-15 
there appears a table of ninety columns (grouped by sixes, however), 
which contains the judgments of ninety judges on some specimens 
of handwriting (page 82). This table will never be read by the public. 
The Birmingham report for 1914-15 contains a table of "Distribu- 
tion of Expenditures" (page 8) which has twenty-six columns not 
broken in any way. Many similar examples might be cited. In all 
probability none of these tables, so far as the average man is con- 
cerned, is worth printing. 

Once in a while a long table of receipts and expenditures 
is justified on the ground that the citizens demand it. 
They may demand it even though they will never read it, 
because they think it will secure greater honesty and 
efficiency in handling school moneys. But such a table 



Presenting Statistics to the Public 211 

may be more tolerable by using double entry as in Table 
22. 

Table 22. Double Entry Table Showing Receipts and Pay- 
ments, Springfield, Illinois, Public Schools, 1912-13 ^ 



Receipts 

State school fund . . . . 
General property taxes . . 
Rent and interest . . . . 

Tuition fees 

Sales of property . . . . 
Sales of equipment . . . . 
Balance at beginning of year 

Total 



Payments 

Board of Education office 
School elections .... 
Finance accounts . . . 
Legal services .... 
Operation of office building 
Office of superintendent . 
Compulsory education . . 

Instruction 

Salaries of supervisors . . 
Salaries of principals . . 
Salaries of teachers 

Textbooks 

Stationery and supplies 
Other expenses .... 

Operation of plant 

Wages of janitors . . . 

Fuel 

Water 

Light and power .... 
Janitors' supplies . . . 





$17,414.19 


— 


314,788.96 


— 


4,176.84 


— 


2,538.50 


— 


2,807.40 


— 


297.28 


— 


264,597.87 


— 


606,621.04 


$2,353.68 




40.00 


— 


720.00 


— 


1,295.50 


— 


730.00 


. — 


3,000.00 


— 


1,000.00 


— 


6,550.00 





30,177.25 


— 


169,012.99 


— 


325.00 


— 


8,914.90 


— 


3,807.25 


— 


16,420.91 





6,519.37 


— 


2,046.08 


■ — 


1,268.05 


— 


1,050.00 


— 



Springfield, Illinois, Survey, p. 99 



212 



School Statistics and Publicity 



Maintenance of plant 

Repair of buildings and care of grounds 

Repair and replacement of equipment . 

Insurance 

Auxiliary agencies 

Libraries 

Promotion of health 

Miscellaneous 

Rent 

Permanent outlays 

Land . . 

New buildings 

Alteration of old buildings .... 

New equipment of new and old build- 
ings 

Other payments 

Redemption of bonds 

Payments of interest 

Total 

Balance at end of year 



Grand Total 



$16,276.98 




5,346.11 


— 


533.10 


— 


850.00 





1,434.00 


— - 


120.00 


— 


9,249.80 


— 


85,527.91 


— 


20,000.00 


— 


2,963.61 


— 


10,500.00 





6,663.75 


— 


414,696.24 





191,924.80 


— 


— 


$606,621.04 



However, although Table 22 gives the classified feature, 
it does not appear on one page and could not be gotten on 
one page in print large enough to be read easily. It would 
make a better showing if printed on adjacent pages, a thing 
easily possible in reports but not feasible here. 

Long tables are also broken into smaller units through 
the use of subdivisions of the main classes of items. The 
question, however, arises as to how far such subdivision 
may extend without making the table too complicated. 
Certainly the process of subdivision should not go farther 
than the description of school buildings in Table 23. 



CO 

o 

O 
1^ 


>^ 
































^^ 
















CO 

o 

O 

o 




CO 
















































1 
















o 

■♦J. 
^ ■ 
















CO 
CO 
















CO 
O 
















Brick or 
frame 
















O 


















O 

o 

t/2 

























o 
o 

CO 



o 



^ 
^ 



02 






213 



214 



School Statistics and Publicity 



Long tables may also be broken up by grouping the items. 

Thus, contrast the tables of Cincinnati and Cleveland for com- 
parison of expenditures for different years, as given by Snedden and 
Allen, pages 35-36, here shown as Tables 24 and 25 : 



Table 24. Example of Effect of Unclassified Items. Com- 
parison OF School Expenditures for the Years 
1895-1905 (Cincinnati) 





Year ending 


Year ending 


Year ending 




Aug. 31, 1895 


Aug. 31, 1900 


Aug. 31, 1905 


Teachers, day schools 


$669,752 


$799,286 


$815,719 


Teachers, night schools . 


9,606 


6,612 


8,321 


Officers and examiners . 


15,143 


16,646 


17,792 


Librarians 








Janitors 








New buildings .... 








Repairs 








Lots 








Furniture 








Heating fixtures . . . 








Rent 








Fuel 








Supplies 








Printing 








Advertising 








Gas 








Census 








Textbooks and supple- 








mentary readers . . 


1,641 


3,502 


13,448 


Incidentals, etc. . . . 


5,678 


4,643 


3,284 


Teachers' Institute . . 








Interest and redemption 








of bonds 








Public library .... 








Deaf-mute taxes . . . 








Transfer of funds . . . 








Apparatus 








Totals 









Presenting Statistics to the Public 215 



Table 25. Example of Effect of Classified Items. Com- 
parison OF School Expenditures (Cleveland) 



August 31 


1900 


1905 


Tuition 

Supervisors' salaries 

Teachers' salaries 

Maintenance 

Officers' and employees' salaries .... 

Fuel and light 


$37,406 
897,190 

118,664 
1,295 


$50,964 
1,314,660 

184,144 


Repairs 




Stationery and supplies 

Contingent 

"Water 


10,335 


Fixed charges 

Interest 




Bonds 




Rent and insurance 

Furniture and fixtures 

Permanent improvements 

Land 




Buildings 




Grading, paving, etc 

Improvement on existing buildings . . . 

Miscellaneous 

School books 




St. Louis Exposition 

Glenville annexation 




Total 











216 School Statistics and Publicity 

4. Arrangement of Totals So That They Can Be Quickly 

Grasped 

This will aid the superintendent to discover significant 
relations, as well. The most common devices are the 
use of the word '' total," the employment of black-face 
type or italics, and the outset in another right-hand 
column ; thus : 

$ 673 $ 673 $ 673 

1240 1240 1240 

890 890 890 $2803 

Total. $ 2803 $ 2803 

5. Printing of Headings So They Can All Be Read from 

One Way 

It is not pleasant to have to twist one's head or turn 
the book around to read vertical headings. The average 
reader will skip such rather than do the turning. For 
the same reason, tables that are printed lengthwise on 
the page should be avoided if at all possible. In general, 
the headings may, by the use of syllables and abbre- 
viations, be horizontally printed in just as small a space 
as the wrong way takes. See pages 72, 213. 

6. Filling in Tables Only Where Data Actually Exist 

If vacant places are filled with zeros, the labor involved 
in reading the zeros is as heavy as that in reading the 
actual numbers.^ However, if the paper is not cross- 
lined, it is better to fill the vacant spaces with dotted 
lines, as the reader may get lost if several vacancies occur 
together. Compare for example Tables 26 and 27 2. 

1 This does not in any way discount what was said on page 84 about 
making an entry for every case. That referred to a blank for the 
superintendent to use in collecting data, not for the public. 

2 Snedden and Allen : School Reports and School Efficiency, p. 78 



Presenting Statistics to the Public 217 



Table 26. Showing Use of Zeros — Detroit Central 

High School 

Total number of " first year " pupils who have left since Sept., 1904 



Cause 



Illness 

Illness in family 
Failing eyesight 

Work 

Transferred . . . 
Left city .... 
Indifference to work 

Music 

Unknown . . . 

Total .... 



Ages 



13 



Bi G T 



13 



B G T 



14 



B G T 



15 



B G T 



Totals 



B G T 



65 



23 
4 
2 

41 
6 

10 
9 
1 

23 

119 



Table 27. Omitting Zeros — Detroit Central High 

School 



Total number of "first 


year 


' pupils who have left 


since 


Sept. 


, 1904 






Ages 


Totals 


Cause 


13 


13 


14 


15 






B 


G 


T 


B 


G 


T 


B 


G 


T 


B 


G 


T 


B 


G 

18 
4 
1 
7 
2 
7 
1 
1 
13 
54 


T 


Illness 


1 




1 


2 


1 


1 

2 


1 

1 

5 


1 

1 

1 
1 

1 


2 

1 
6 
1 
1 

1 


1 

10 
2 

2 

4 


4 
1 
1 
2 
1 
3 

1 
6 


5 
1 
1 

12 
3 
3 
2 
1 

10 


5 

1 

34 

4 

3 

8 

10 
65 


23 


Illness in family . . . 
Failing eyesight . . . 

Work • . . 

Transferred 

Left city 


4 
2 

41 
6 

10 


Indifference to work . . 

Music . 

Unknown 

Total 


9 

1 

23 

119 







^ In order to include all the columns within the limits of the page, it has 
been necessary to use the abbreviation B for Boys, G for Girls, and T for 
Total. 



218 School Statistics and Publicity 

7. Care to Avoid Eyestrain 

All the type in a table should be large enough to be 
read easily and quickly. It is a false economy to compress 
tables into a very small space through the use of fine print. 
This may save money on printing bills. But it loses 
more money, because the fine print will not be read by the 
very persons for whom the table was prepared. 

Dotted lines are very helpful to aid the eye in covering 
long horizontal stretches between data that are to be 
connected. They should, however, go as far as they are 
needed. 

In Table 28 dotted lines start out, but do not go far enough to be 
of real service. 

Table 28. Example of Bad Use of Dotted Lines. Cost for 
Overhead Administrative Control in Western Cities ^ 

City Percent of total mainte- 

nance cost spent for ad- 
ministrative control 

Sacramento, Cal 1.8 

Spokane, Wash 2.2 

Pasadena, Cal 2.4 

Seattle, Wash 2.6 

Oakland, Cal 2.7 

Denver, Colo 2.7 

etc. etc. 

8. Avoidance of the Alternate Column Scheme Unless 
the Types and Arrangement in the Columns Stand Out 

Sharply 

For example, it is usually unwise to use alternate lines 
for statistics on boys and girls, for public consumption, 
unless one set is in red type, black-face type, or italics. 
Italics as used by Ayres in his ** Laggards in Our Schools " 

1 Oakland Survey, p. 17 



I 



Presenting Statistics to the Public 219 

are generally less satisfactory than black-face type. As 
red, for practical purposes, can be used only on individual 
copies and not in printed reports, it is often better to 
make a separate table for boys and one for girls if they 
are to be printed. This simplifies the matter. Both 
tables may be later summarized into one, as on page 217, 
for purposes of comparison which will be clear to any one 
who has looked at the previous tables. 

In the Des Moines report for 1914-15, a table of this kind appears, 
opposite page 98. It is an age-progress table. Children who have 
always been in school in Des Moines are shown in the upper left-hand 
column of each rectangle; those coming in, in italics in the upper 
right-hand column ; and the total below in heavy type : 



20 SU 
54 



The differentiation in lines within divisions of a table 
must also be carefully attended to. Note in Tables 26 
and 27, page 217, how lighter lines are used to subdivide 
the larger divisions. Dotted lines would have done as 
well for the interior lines, but the use of colons for this 
purpose is not advisable. 

The Salt Lake City Survey has a table ^ in which colons have been 
used to indicate interior lines. But as set up, they appear at first 
to indicate ratios between boys per seat and the percentage of suffi- 
ciency, and so on. It takes the average reader some time to realize 
that these colons are really intended to form dotted dividing lines. 

9. Clear Headings and Subheadings 

The subheadings must be clear without much explana- 
tion. The use of only key numbers or letters to head 
columns is bad enough for the person doing the work.^ 

1 Page 249 2 gee page 74 



220 School Statistics and Publicity 

Every effort should be made to avoid it in a tabulation 
intended for the average man. 

On page 50 of the Hammond Survey appears a table employing key- 
letters which actually occupy almost as much space for explanation 
as the tabulation itself. 

10. Neatness and Artistic Features 

In addition to the foregoing, it is highly desirable to 
have all printed tables as neat and artistic as possible. 
Tables that are pleasing to the eye will by their very 
form and convenience attract unconscious attention, 
which may then easily be diverted to a consideration of 
the ideas or conclusions embodied, in the tabulated 
material. Or they will cause the reader so little strain 
that whatever attention he has to give them will be 
concentrated on the thought alone. A competent printer 
will be able to secure such results by himself but may 
need to be held down to setting up all type so that it can 
be read from one position. Where the superintendent 
has to give complete directions for printing, he should be 
especially careful about uniform headings, uniform spac- 
ing, large print, inclosing each table in a border, or at 
least using distinctive ruling or open spaces above and 
below each table to set it off properly. The tables in 
this book have been planned to serve as models on these 
matters. 

III. HOW TO MAKE UP A SERIES OF TABLES OF THE 
SAME GENERAL NATURE 

Often it is desired to have a series of tables of the 
same general nature or at least closely related. This 
necessitates : 



I 



Presenting Statistics to the Public 221 

1. A Summary Table at the Start and Minor Tables in 
the Same Sequence as Items in the Summary Table 

For example, if it is desired to give a series of tables 
on various school costs per pupil, the procedure might 
be thus : First, have a summary table with various items, 
as cost of superintendent's office, cost of instruction, 
cost of supervision, etc., all appearing in the same table 
for the city as a whole. This could be followed by a 
series of tables, each covering all the ward schools on one 
item. These tables would follow the sequence of the 
items in the summary table. 

2. The Same Sequence of Items within Similar Tables 

If, in these various tables, the lump sum, the average 
number of pupils belonging, and the cost per pupil are 
given in adjacent columns from left to right, it is advisable 
to follow the same order of presentation and the same form 
of table all the way through. In this way the " mind 
set " of the reader may be utilized. 

3. Keeping in Mind That the Main Purpose of Any 

Tabulation Is the Showing of Relationships 

(a) The purely alphabetical order of items in many tables 
destroys or greatly handicaps the showing of relationships. 

Thus in the report of the city superintendent of South Bend for 
1913-14, all the tables involving the different wards or schools are 
practically worthless for influencing the public because of this alpha- 
betical arrangement. For example, take the table on comparative 
costs of instruction and supervision by buildings for 1913-14, page 30. 
If the last column of this table had been arranged in order of magni- 
tude from high to low, the comparison would stand out. See Tables 
29 and 30 for this. 

The original table might do if the central tendency and quartiles 
for the city, or some other standards, were printed in bold-faced type 



222 



School Statistics and Publicity 



at the top of the table, so that comparison might be made with them 
for each ward school. Or it would not be so unsatisfactory if another 
column had been added at the right giving the rank of each school 
on cost per pupil, from highest to lowest. Grammar would be rank 
2; Colfax, rank 9; and so on to Warren, which would be rank 3. 
But the ranks would not show up so well as in the arrangement on 
page 223. In any event, it is only just to note that the alphabetical 
arrangement is of service in making easy the work of checking the 
names of the schools so that no one will be omitted. This has its 
value during the period of working up data inside the school system. 
But this value disappears as soon as the table reaches the citizen. He 
assumes that the work is correct and desires only to get at the meaning 
of the whole and its parts as quickly as possible. 

ORIGINAL FORM 
Table 29. Example of Alphabetical Arrangement of Items 
Showing Comparative Cost of Instruction and Super- 
vision BY Buildings for 1912-13 (South Bend) 





Total 


Average 


Cost 


Schools 


cost of 


daily 


per 




instruction 


attendance 


pupil 


Grammar 


$10,054.05 


332 


$30.28 


Colfax 


etc. 


etc. 


24.53 


Coquillard 






25.36 


Elder 






22.19 


Franklin 






21.04 


Jefferson 






32.35 


Kaley 






22.26 


Lafayette 






27.97 


Laurel 






20.14 


Lincoln 






24.07 


Linden 






25.10 


Madison 






26.12 


Muessel 






27.94 


Oliver 






19.90 


Perley 






23.61 


River Park ....... 






19.25 


Studebaker 






23.89 


Warren 






29.16 



Presenting Statistics to the Public 223 

REVISED FORM 

Table 30. Bobbitt Table Arrangement of Data in 

Table 29 

Schools Cost per pupil 

Jefferson $32.35 

Grammar 30.28 

Warren 29.16 

Lafayette . . . . . 27.97 

Muessel 27.94 

Madison 26.12 

Coquillard 25.36 The full figures from which 

Linden 25.10 this table is derived are on 

Colfax ....... 24.53 file in the superintendent's office 

Lincoln 24.07 and may be inspected by any 

Studebaker 23.89 interested person. 

Perley ...... 23.61 

Kaley 22.26 

Elder 22.19 

Franklin 21.04 

Laurel 20.14 

Oliver 19.90 

River Park 19.25 

(6) It is generally best to arrange the table in the form 
of a scale running from high to low. 

There are some exceptions. Perhaps a better rule 
would be to arrange items in the table so that the city or 
school having the desired trait in the greatest abundance 
should be at the top. 

For example, a table showing the number of dollars behind each $1 
spent for schools should be placed with the smallest number at the 
top. The smaller number is the more desirable, since a small number 
of dollars behind each dollar spent for schools indicates a high school 
tax and presumably good schools. Accordingly, Table 31 should 
have been reversed. 



224 School Statistics and Publicity 

Table 31. Real Wealth Behind Each Dollar Spent for 
School Maintenance ' 

1. Atlanta, Ga $559.00 

2. Los Angeles, Cal 538.00 

3. Richmond, Va 536.00 

4. Birmingham, Ala 479.00 

5. Portland, Ore 456.00 

6. Memphis, Tenn 449.00 

7. Indianapolis, Ind 408.00 

and so on to 

35. Toledo, Ohio 184.00 

36. Worcester, Mass 180.00 

37. Newark, N. J 165.00 

Note that the ranks in this are really reversed. Atlanta in this 
showing is doing less for schools than any of the other cities and should 
have rank 37. Newark should have rank 1. 

Table 32, which gives the same idea in another way, is correct in 
putting the largest number at the top, because a high tax rate on real 
wealth for schools is desirable. It is given this way : 

Table 32. Comparative Rates of Tax Required for School 
Maintenance (in Mills) 2 Based on Real Wealth of Cities ^ 

1. Newark, N. J 00606 

2. Toledo, Ohio 00543 

3. New Haven, Conn 00541 

4. Paterson, N. J. 00541 

5. Lowell, Mass 00515 

and so on to 

31. St. Paul, Minn 00244 

32. Memphis, Tenn 00244 

33. Portland, Ore 00219 

34. Birmingham, Ala 00209 

35. Richmond, Va 00186 

36. Los Angeles, Cal 00184 

37. Atlanta, Ga 00180 

1 Portland Survey, p. 310 

2 In some parts of the country, this would be more easily under- 
stood if given as cents on the one hundred dollars ; as : 

1. Newark, N. J $.606 

^Portland Survey, p. 311 



Presenting Statistics to the Public 225 

In printing Bobbitt tables, the most effective results 
on the casual reader will undoubtedly be obtained by 
giving each item a separate line. 

Thus in Tables 33-35 from pages 62 and 63 of the San Antonio 
Survey, Form 1 is best, Form 2, the next best. Form 3 should not be 
used. A trained reader can understand one form as easily as the 
other. But the average man will understand the first form much 
more quickly than he will the others. 



Table 33. Bobbitt Table, Form 1. Annual Per Capita 
Expenditures for Street Maintenance, 1912 

Nashville $2.79 

Augusta 2.76 

Tampa 2.10 

Memphis 2.04 

Houston 2.04 

Savannah 1.71 

Atlanta , 1.63 

Dallas 1.55 

Galveston 1.54 

Jacksonville 1.53 

Austin 1.51 

New Orleans 1.50 

Macon 1.43 

Shreveport ........ 1.36 

Montgomery 1.36 

Mobile 1.33 

Fort Worth 1.17 

El Paso 1.10 

Muskogee 1.07 

Birmingham 1.02 

San Antonio 99 

Charleston 85 

Little Rock ........ .63 

Oklahoma City ..,.,.. .63 



226 



School Statistics and Publicity 



Table 34. Bobbitt Table, Form 2. Annual Per Capita 
Expenditures for Street Maintenance, 1912 

Nashville $2.79 

Augusta 2.76 

Tampa 2.10 

Memphis 2.04 

Houston 2.04 

Savannah 1.71 

Atlanta 1.63 

Dallas 1.55 

Galveston . . . . . 1.54 

Jacksonville 1.53 

Austin 1.51 

New Orleans . . . . 1.50 



Macon . . . 






$1.43 


Shreveport . . 






1.36 


Montgomery . 






1.36 


Mobile . . . 






1.33 


Fort Worth . . 






1.17 


El Paso . . . 






1.10 


Muskogee . . 






1.07 


Birmingham 






1.02 


San Antonio 






.99 


Charleston . . 






.85 


Little Rock . . 






.63 


Oklahoma City 






.63 



Table 35. Bobbitt Table, Form 3 (Practically Never De- 
sirable). Annual Per Capita Expenditures for Street 
Maintenance, 1912 



Nashville 

Tampa 

Houston 



$2.79 
2.10 
2.04 



Augusta . 
Memphis 
Savannah 



$2.76 
2.04 
1.71 



Atlanta . 
Galveston 
Austin . . 



1.63 
1.54 
1.51 



Dallas . . 
Jacksonville 
New Orleans 



1.55 
1.53 
1.50 



Macon . . 
Montgomery 
Fort Worth . 



1.43 
1.36 
1.17 



Shreveport 
Mobile . 
El Paso . 



1.36 
1.33 
1.10 



Muskogee 
San Antonio 
Little Rock 



1.07 
.99 
.63 



Birmingham 
Charleston . . 
Oklahoma City 



1.02 
.85 
.63 



The use of Forms 2 and 3 probably grows out of a desire to utilize 
part of a page for the table, or through a mistaken idea that it econ- 
omizes space. But in most places where it is desired to use such a 
table, there would be many such tables to present. It is readily 
apparent that two tables of Form 1 placed side by side on a page 
would take up practically no more space than if they were printed 
either as Form 2 or Form 3. 



Presenting Statistics to the Public 227 

(c) If many horizontal lines appear in the table, ease 
in reading them is facilitated by running a heavy horizontal 
line or leaving a space every five lines or less. 

Examples of this are familiar to most readers. The lines drawn 
across, or the gaps left in the Bobbitt tables to show the medians and 
quartiles serve the same purpose, if there are not too many cases in 
each section of the distribution. A third device for this purpose is 
the numbering of the items from top to bottom on both margins. 
Thus the data for item 3 on the left may be traced across to the data 
on the line numbered 3 on the right. This is the device used by the 
United States Bureau of Education on many of the tables that cover 
two pages, with continuous horizontal lines. 

(d) In most tables where the cases are kept separate, 
it is advisable to make the name of the city, school, etc., 
to be compared with the others, stand out prominently. 

It will be recalled that this was done in the tables from the Port- 
land Survey. The device is the familiar one used by newspapers to 
call attention to the home city in a table giving the standing of the 
baseball clubs of a league. Sometimes the emphasis is given with 
capital letters as in Portland, or with black-faced type as in Salt Lake 
City. On paper charts not to be printed, the particular case can be 
marked in red or some bright color. It is hardly possible to make the 
one case and the data with it stand out too prominently. 

In the age-grade tables, it is customary to make the children of 
normal age stand out prominently by marking off these numbers. 
Heavy stairstep lines may be drawn to inclose the numbers for nor- 
mal children; or heavy lines may be placed above and below these 
numbers ; or they may be boxed in ; or they may be printed in bold- 
faced type. Samples are given in Figure 29. 

Where the printer can handle it, the boxed-in form is preferable, 
because it enables one to separate the normal from both the retarded 
and the accelerated children very easily. 

In another form of age-grade table, the facts for one grade at a 
time are presented. This is usually done by running the ages along 
the top and the number of years the child has been in school down the 
table. Then two heavy lines are drawn vertically through the table 
to inclose the children of normal age in this grade. Two other heavy 



228 



School Statistics and Publicity 









Ag 


as 






Grades 


under 
6 


6 

to 

6k 


TO 
7 


7 

to 

7i. 


to 

8 




IB 


22 1 54-0 


)59 1 105 


38 




1 A 




30 


98 


il8 


76 




HB 




17 


6^ 1 133 


102 




HA 






Z 


£7 


58 


e.. 


HIB 














HEA 















Ages 



Grades 

I B 
lA 

EB 
BA 
IHB 
HIA 



under 
6 


6 
to 


to 

7 


7 

to 

7i 


7i 

to 

8 




22 


540 


159 


105 


38 






30 


98 


M8 


76 






17 


65 


133 


102 








a 


Z7 


58 


84 



























Ages 



under 
6 


6 

to 


6i 

to 

7 


7 . 

7i 


7i 

to 

8 




ZZ 


540 


159 


105 


38 






30 


98 


lie 


76 






17 


65 


133 


lOZ 








a 


27 


58 


e4 



























Fig. 29. 



Grades 

IB 

I A 

HB 

HA 
niB 

HLK 



Devices for Making the Number of Normal Children Stand 
Out in Age-Grade Tables. 



Presenting Statistics to the Public 229 

lines are drawn horizontally through to inclose the children of normal 
progress irrespective of how old they were at entrance. The area 
in the center, where the two sets of lines cross, incloses children who 
are both normal for age and progress. Various other combinations 
for the other parts of the table are easily worked out. 

The following table form is similar to one used in the Bridgeport, 
Connecticut, Survey, page 35. 



Years 

in 
School. 



Table 36. Age-Progress Table. 5B Grade 



Ages. 



Total. 





6 


6i 


9 


3k 


10 


I0| 


II 


Mi 


12 


»2i 


13 




2 


























2k 


























3 


























3^ 


























4 


























^i 


























5 


























5i 


























6 


























6i 


























7 


























Total. 



























(e) It is often advisable to present only percentage 
derivations instead of the original figures. 



230 School Statistics and Publicity 

The reason is that the original figures may be large and not easily 
comparable, whereas the percentage equivalents are small and easily 
compared. Distribution tables of the results on standard tests, how- 
ever valuable for educational investigations, are not at all suited for 
the general public. Round numbers or the nearest whole units are often 
better than exact figures, as they can be held in mind more easily. 
If all the parts of 100 per cent are shown, care should be taken to as- 
sign values to the parts that will total exactly 100 per cent. Approxi- 
mations are satisfactory for the superintendent, but a discrepancy be- 
tween the sum of the parts and 100 per cent would afford some readers 
an excuse for attacking the accuracy of the report. These suggestions 
must, however, be used with caution. For in some instances the use 
of approximate figures may lead readers to suspect that the figures 
have been " doctored." Especially is this true as regards statistical 
presentations designed to increase the school tax. 

(/) Special devices in tabulating are sometimes of great 
value. 

For example, a tabulation showing the part of each dollar of school 
money that goes to various school expenses is effective with the average 
man. Insurance companies take great pains to show where every part 
of the dollar from premiums goes. Tables 37-39 show some of the 
possibilities. Similar devices could show the part of a year or years 
spent on each subject. 

Table 37. How Portland Spends Its Dollar ^ 

Interest 20.7 cents 

General expenses of city government 6.0 cents 

PoHce department 9-0 cents 

Fire department 12.0 cents 

Inspection 0.9 cents 

Health 0.7 cents 

Street cleaning and sanitation 6.2 cents 

Care of streets and bridges 9.0 cents 

Education 30.8 cents 

Libraries 1.0 cents 

Parks and playgrounds 2.6 cents 

Damages 1-1 cents 

Total lOO.O cents 

1 Adapted from the Portland Survey Graph, p. 84 



Presenting Statistics to the Public 231 

Table 38. How the Trolley Nickel is Divided ^ 

2.02 cents for wages 

.78 cents for expenses 

.55 cents for taxes 
1.06 cents for interest 

.59 cents for dividend 
5.00 cents — your nickel 

Table 39. How Rockford Spent Its School Dollar, 1915 ^ 

Elementary Schools 

Teachers' salaries 41.67 cents 

Building maintenance and upkeep . . 12.3 

New buildings ........ 9.9 

Educational supplies 3.9 

Department of hygiene .97 68.74 cents 

General 

Interest on school fund 2.81 

Executive employees (Educational) . 1.39 
Executive employees (Board) ... .86 
Evening schools and gymnasium . . .4 5.46 

High School 

Teachers' salaries 15.81 

Upkeep of building 6.7 

Educational supplies 1.43 

New building 1.41 

Educational employees .45 25.80 

100.00 cents 

Sometimes the variations in sizes of different items may be in- 
dicated by difference in sizes of the type used in printing. Table 40 
is a good example of this, but the variations in size are only very 
rough approximations. 

1 Poster used by Louisville, Ky., Railway Company 

2 Adapted from the chart on p. 120 of the Review of Rockford Public 
Schools, 1915-16. Some of the figures were changed slightly so as to 
make the total exactly 100 cents. 



232 



School Statistics and Publicity 



Table 40. Variations in Type to Indicate Relative Size 

"Watch the Central Association Grow!" 



Year 


i^ew memoers auucu 
during year 


1 1 uuti paiu up 
membership 


Net increase 


1908 

1909 

1910 ' 

1911 
1912 
1913 
1914 


102 

161 

132 

117 

115 

146 

177 


824 
445 

484 
486 

659 

619 

680 


121 

39 

2 

73 
60 

61 


1915 


223 


768 


88 


1916 


320 


973 


205 


917 


" Every member 

get one I " 


Are you one 
of these ? 


"There is no 
going back." 



Do you need more convincing proof of the worth of membership in 

The Central Association of SCIENCE and MATHEiVIATICS TEACHERS? 



Table 41 uses very little space, for data classified in several ways. 

Table 41. Example of Presenting in a Small Space Data 
Classified in Several Ways. The Standing of Salt Lake 
City in the Fundamentals of Arithmetic as Compared with 
Other Cities, Judged by the Median Score Attained by 
Each Grade ^ 





Addition 








Multiplication 


V 


VI 


VII 


VIII 




V 


VI 


VII 


VIII 


3.9 


4.6 


5.4 


6.7 


Detroit 


3.8 


4.8 


6.0 


7.5 


3.7 


4.9 


5.6 


7.8 


Boston 


3.3 


4.8 


5.1 


6.5 


3.9 


4.4 


4.7 


5.6 


Other Cities 


2.6 


4.5 


5.2 


6.4 


2.9 


3.4 


3.8 


5.3 


Butte 


4.1 


5.0 


6.5 


8.1 


4.1 


6.4 


6.9 


8.6 


Salt Lake City 


4.3 


6.3 


7.1 


8.3 




Subtraction 








Division 




5.5 


6.2 


7.3 


9.5 


Detroit 


2.7 


4.4 


7.1 


8.8 


4.9 


6.3 


6.9 


8.6 


Boston 


2.0 


3.3 


5.1 


6.9 


4.5 


6.1 


7.8 


8.4 


Other Cities 


2.3 


4.3 


5.8 


6.3 


2.9 


3.4 


3.8 


5.3 


Butte 


3.6 


4.3 


7.2 


10.2 


6.2 


7.8 


8.8 


9.8 


Salt Lake City 


3.0 


6.6 


7.7 


9.6 



^ Salt Lake City Survey, p. 174 



Presenting Statistics to the Public 233 

(g) There are times when it is necessary to use con- 
ventional tables. In so doing, the following points may 
be of use: 

1. Use double distribution tables and forms recommended by the 
National Education Association committee.^ 

2. Place dates old to new down the page or left to right. 

3. Place magnitude so that the most desirable showing is at the 
top or to the left. (This sometimes reverses the order of dates ad- 
vocated in 2.) 

4. Use Roman numerals for one classification, e.g., the numbers 
for the school grades, and Arabic numerals for the data on pupils 
within the grades. 

5. It is best where the sense is not destroyed and the effect de- 
sired will not be lost, to get up tables in the conventional forms. 

Bizarre effects in tabulation are no more to be desired than is 
writing from right to left or up the page. 

EXERCISE 

Take some annual school report or school survey in which you 
are interested. Write out a detailed criticism of the tables or lack of 
them in it, from the standpoint of their effectiveness with the public, 
showing just why they are good or liable to be unsuccessful. In the 
cases of the unsuccessful ones, or failure to employ tabulations where 
desirable, draw up forms that would present the same data properly. 

REFERENCES FOR SUPPLEMENTARY READING 

Report of the Committee on Uniform Records and Reports. U. S. 

Bureau of Education Bulletin, 1912, No. 3. 
Rugg, H. 0. Statistical Methods Applied to Education, Chapter X. 
Snedden, David S., and Allen, William H. School Reports and School 

Efficiency. 

1 Report of Committee on Uniform Records and Reports, Bureau 
of Education Bulletin, No. 2, 1912, p. 20 



CHAPTER XI 

GRAPHIC PRESENTATIONS OF SCHOOL STATIS- 
TICS, ESPECIALLY FOR THE PUBLIC 

I. OBJECT OF GRAPHIC PRESENTATIONS 

The object in making a graphic presentation of statis- 
tical matter is to give as quickly as possible through the 
eye a faithful and forceful bird's-eye view of the mass of 
statistics, the significant parts, their relationships, etc. 
Graphic presentation in statistics is simply a develop- 
ment of a growing general tendency to make desired 
impressions by pictures rather than by word descriptions. 
This tendency is shown most clearly in the pictorial 
supplements and cartoons, and in the constantly in- 
creasing proportion of illustrations in printed material 
appearing in all our leading newspapers and magazines 
at the present time. 

The graph, however, is more closely akin to the line 
drawing than it is to the photograph. It is found to be 
of great advantage in textbooks for the following reasons : 

1. It presents the significant points in a clear and unmistakable 
way. 

These points are also presented apart from the great mass 
of subsidiary data on which they rest. 

2. It makes the presentation concrete by appealing to the eye. 

As many people are unable to understand things they 
cannot image, the graph will drive the significant points home 
to those who could not be reached otherwise. 

234 



Graphic Presentations of School Statistics 235 

3. It often economizes time and space, for it will take up less room 
than the description it displaces. 

4. It gives to most persons a more accurate basis of comparison 
than they could get with the same effort from word descriptions or 
tabulations. 

Textbooks must make things clear very quickly to 
beginners in the subject. School reports must present 
school facts very rapidly and clearly to citizens who 
know little of them. Consequently, the graph, if prop- 
erly used, should be about as valuable for school reports 
as it has been for textbooks. 

The graph, however, is open to dangers of mis- 
representation and exaggeration. These things are just 
as harmful in school graphs as they are in demagogic 
politics, patent medicine advertisements, etc. The ideal 
graph would probably have the forcefulness of the most 
powerful motor car, department store, or patent medicine 
advertisements, with something like the truthfulness 
and accuracy on essential points of a first-rate scientist. 
Another trouble is that a graph leaves with most persons 
only a general impression ; it is very difficult for them to 
recall it accurately, much less to reproduce it from memory 
later for any one else. 

II. HOW TO MAKE GRAPHS FOR THE PUBLIC FROM 

STATISTICAL DATA 

1. Component Parts 

Circle Graph. It is often desirable to show in a graph 
the relative size of the component parts of a statistical 
whole. A popular device for doing this is the circle, 
each sector by its area representing one proportional part 
of the whole. 



236 School Statistics and Publicity 

Figure 30 is a graph of this nature showing the distribution of time 
through the eight grades for the various common school subjects. 
Figure 31 gives a similar graph with fewer parts. 




Fig. 30. — Component Part Circle Graph Showing Distribution of Time 
to Various Subjects Throughout the Eight Elementary Grades. 

(Adapted from the 1915-16 Review of the Rockford, Illinois, Schools, page 52.) 

This kind of graph is famihar to the public and of 
course makes the comparisons through the sizes of the 
angles. But it has these disadvantages : 

1. If more than a few parts are to be represented, there is trouble 
in reading the names. 



i 



r 



I 



Graphic Presentations of School Statistics 237 

When there are only three parts and they are all large, the 
names may all be printed horizontally as in Figure 31. But 
if there are many parts and some of them are small, this is im- 
possible. Then the names 
must be printed as in Figure 
30 or else at the side with 
dotted lines leading into the 
parts indicated. If the 
former device is used, the 
printing must be reversed 
as the eye proceeds around 
the circle, with consequent 
delay. Note in Figure 30 
how one has to read up 
for "music" and down for 
" geography." If the latter 
device is used, much more 
time is consumed in asso- 
ciating the names with the 
parts concerned, than when 
the names are printed on 
the parts. 




Fig. 31. — Component Part Circle 
Graph Showing Relative Propor- 
tions of Normal, Retarded, and 
Accelerated Pupils. 

(From Salt Lake City Survey, page 190.) 



2. The figures denoting the various parts cannot be placed in such 
positions that they can be easily compared or added. This is espe- 
cially bad when the parts are given in percentages. 

3. It is extremely difficult to compare the same factors in different 
wholes. 

For example, suppose a circle similar to this one for Rockford 
" had been drawn for Joliet; it would be very hard to com- 

pare the sector on "reading" in the Rockford circle with the 
' corresponding sector in the Joliet circle. 

Bar Graph. The data shown in this circle graph for 
Rockford may be presented in a bar graph which permits 
of placing the figures so they can be added. For this 
the parts within each of the three main divisions should 
be arranged from high to low. Figure 32 shows this 
arrangement. 



I 



I 



238 



School Statistics and Publicity 



Reading 



Arithmetic 



Note that practically all the defects of the circle graph have been 
remedied here. Note also that the component parts in a bar graph 
are shown proportionately by their lengths only. Their widths have 

nothing to do with it and their 
areas are in exactly the same 
proportion as their lengths. 

Note, too, that the figures arc 
given for any reader who cares 
for them and in such shape that 
they may be added easily. 

Sometimes for economy in 
printing, the bar may be a hori- 
zontal one, in which case the 
lines of printing and figures may 
run vertically so that the reader 
will have to turn the page to 
read them. 

English 12.86% . i i ^. • ., 

An elaboration of the 

component bar graph is 

the form with two different 

but related scales, one on 

either side. 

For example, the waste 
through repeaters in a school 
system might be shown as in 
Figure 33, using the average 
annual cost per elementary pupil, 
say $30. 

The bar graph idea with 
component parts may be 
100 7o carried out for popular 

Fig. 32. — Bar Graph Showing by Presentation with cartoon 

Component Parts with Subdivisions effccts. 
the Distribution of Time for Com- 

mon School Subjects. Po, example, the proportion of 

(Adapted from Figure 30.) white and negro children out of 



Geoqrophy 



Hisloi 
Music 



7 



Drawing 

Manual Jr. 
Physical Jr. 

Gomes 
Recess 

Opening Ex. 



t.b% 




14.75 % 




12.867. 




7.64% 




3.09% 




5.85% 




6.65% 




2.85% 




3.267. 




1.09 7o 




706% 




2.94 7o 





Graphic Presentations of School Statistics 239 



No. of 
repeaters 



50 
Z5 
50 

75 

50 
50 

50 

75 
425 



8th Grad^ 


7th 


M 


6th 


II 


5th 


• 1 


4 th 


II 


3rd 


II 


znd 


•t 


1st 


M 




« 



Cost of repeat-' 
ing work 



H500 

750 

1500 

a200 

1500 
1500 

1500 

2200 
,^2750 



Fig. 33. — Bar Graph to Show Component Parts, with Two Different but 

Related Scales. 

each twelve may be shown as in Figure 34. The black children com- 
ing in from the right on the row serve the same purpose as putting 
the right end of a bar in black. But the idea of the children's figures 




Fig. 34. — Bar Graph with Cartoon Effect Showing Proportion of White 
and Colored Children out of Every Twelve in Alabama. 

(From An Educational Survey of Three Counties in Alabama, page IS.) 



240 School Statistics and Publicity 

here gives an interesting touch in addition. While this cartoon 
graph is not accurate, strictly speaking, a little care will make it 
accurate enough for its purpose. For example, by choosing twelve 
rather than ten figures for the whole line, fractions of two-thirds 
and one-third have been shown with whole figures. This device 
is extremely effective in a popular presentation. 

2. Simple Comparisons 

The simplest graphic comparison is usually made 
through some form of the bar graph. The latter is used 
a great deal for comparisons in presentations intended 
for the public, but its possibihties are not generally 
recognized. Many graphic comparisons for pubhc con- 
sumption so far have involved one or both of two 
errors : 

(a) The comparison is made by using similar areas. 

(6) The comparison is made through a cartoon effect, when the 
latter is needed only to attract attention to the graph, in which bars 
would show the relationship much better. 

We shall now examine the possibihties of the bar 
graph for comparisons, and after that take up typical 
graphs of other kinds for this purpose. 

Comparisons with Bar Graphs. Some of the possi- 
bihties of the bar graph in making comparisons have 
been indicated heretofore, particularly in connection 
with a distribution arranged as a Bobbitt table. ^ If the 
component parts of any bar graph on pages 238 and 239 
were taken out, arranged in order of size and lined up at 
the left, they would give a bar graph effect running from 
high to low. 

In making comparisons with bars, it is generally advis- 

1 See p. 103 



Graphic Presentations of School Statistics 241 



able to follow this order, with the bars lined up at the left 
ends, as in Figure 35. 

Name of item Size Bar 



$703 
5,52 




Newark 
Jersey City 

Fig. 35. — Illustration of Correct Order of Items in a Bar Graph. 



By doing this the magnitudes will come in a column where 
they can be seen by those who like to have the figures, 
and these can easily be added, with the sum at the bottom 
of the middle column. Very seldom should the numbers 
be placed at the immediate right-hand ends of the bars. 
They tend to make the bars seem disproportionately 
larger and they cannot be easily compared or added. 



lo 20 30 40 50 60 70 SO 90 <00 



1.- Vermont 
etCy to 
^QrLoutsiana 



m. 




Fig. 36. — Device Used by Dr. L. P. Ayres to Show by States Percentages 
of the School Population Enrolled in Public Schools, in Private Schools, 
and Not in Any School, in 1910. 

White portion indicates children in public schools ; shaded, those in private schools ; 
and black, those not in any school. (Adapted from A Comparative Study of the Public 
School Systems in the Forty-eight States and reproduced by permission of the Russell 
Sage Foundation.) 

To make the bars stand out clearly, the distance between 
them must be markedly different from the width of a 
bar. Otherwise the lines will tend to run together or 
sink into the background. 

•Sometimes, to economize space, some of the largest 
items may be shown with double bars. A familiar example 
is the graph used by the American Book Company on its 



242 



School Statistics and Publicity 



calendar. On this a long double bar represents the 
expenditures in liquor and a short single bar, the expendi- 
tures in textbooks. This is very doubtful practice, as it 
greatly lessens the difference between the magnitudes for 
the untrained reader. 

The horizontal bar graph is also very valuable where 
it is desirable to make comparisons between component 
parts in similar wholes. For this purpose, the wholes 
are represented by bars of the same width and length. 

Republican V//A Progressive ES3 Democrat I I 

AO 50 60 70 80 90 100 



10 20 30 

Alabama J^yyAwww"^ 

to 



New York 




Fig. 37. — Device to Show Parts of a Total and Also to Indicate Relative 

Sizes of the Totals. 

Adapted from Professor Irving Fisher's chart showing parts of the total vote for 
president in 1912, at the same time indicating the relative voting strength of each state. 

Then the component parts for the same item are lined up on the 
left margin, and those for another item will appear lined up on the 
right margin. A fine example is shown in Figure 36. 

It is to be noted that this will not work well for more than two 
component parts. Observe how difficult it is to get an idea of the 
size of the middle item representing the number of children enrolled 
in private schools when this item for different states is compared. 

If, for any reason, it is desirable to make comparisons 
between the relative sizes of the wholes, the widths of 
the bar may be adjusted accordingly. 

Mr. W. C. Brinton borrows such a device for giving an 
analysis of the total vote for president in 1912 in the 
forty-eight states from Professor Irving Fisher.^ Each 

1 Brinton, W. C. : Graphic Methods cf Presenting Facts, p. 10 



Graphic Presentations of School Statistics 243 

state is represented by the same length, and the percentage 
given to each party varies, while the number of voters 
also varies in the states. (See Figure 37.) 

The following are some of the places in school work 
where a graph of this kind would be helpful : 



1. It could be used in the Ayres graph given before for comparing 
the total number of school children in the states. 

2. In a chart comparing the number of retarded, normal, and ac- 
celerated children in a number of cities, the relative totals of children 
enrolled could be shown by the width of the bars. 

100 100 
V 



21 



82 



68 



r: 



^ 



55 



35 








38 














! 






23 

1 






19 

i 


28 




12 12 




1 





r^y^~^ 



13 14 15 16 17 18 19 

Fig. 38. — Bar Graph for Comparing Two Things Whose Proportions Are 

Constantly Varying. 

Columns represent number of boys and girls among each hundred beginners who 
remain in school at each age from 13 to 19. Shaded columns represent boys and white 
columns girls. (From Springfield Survey, page 52, by permission of the Russell Sage 
Foundation.) 



244 



School Statistics and Publicity 



3. If the distribution of the cost of educating one child were being 
shown for various items such as instruction, supplies, etc., for several 
cities, the width of the bar might represent the total amount spent 
by each city to educate one of its children, etc. 

The bar graph may be used to compare two or more 
things whose proportions are constantly varying, by using 
a different shading for each separate kind of item. 

Thus Dr. Ayres uses Figure 38 to show the number of beginners 
that remain in school at various ages at Springfield, Illinois, using 
shading for boys and white for girls. The reader not only gets a 
comparison between the number of boys and girls remaining in school 
at any given age, but also a comparison between the different age groups. 

Other examples are : 

1. Enrollment of pupils in different grades, white columns for boys 
and black columns for girls [Cleveland Summary Volume, page 85). 

2. Contrast in percentage of retardation of white and colored 
pupils by grades, shaded bars for white, black bars for colored pupils, 
all on horizontal basis {Louisville Report for 1911^-15, page 27). 



School. 

year 



Daily 
cost 



Hi<3h 
schools 



Sal 



ones 



1 


'////y 


y//A y////////, y////////.w^^ 


// /// 


y///, mrn^M, '/////////. yrn^M, 


^^^ WMM/M ^vy/////A 





I WASH I NQ TON 
12 VtRMONT 
24 KANSAS 
36 MARYLAND 
48 ALABAMA 

Fig. 39. — Device for Showing Relative Standing of Several Cases on 

Each of Several Items. 
Adapted from Ayres's graph showing the standing of the forty-eight states, by per- 
mission of the Russell Sage Foundation. The highest quarter is represented by white, 
the second quarter by light shading, the third quarter by dark shading, and the lowest 
quarter by black. 

An elaborate yet easily understood chart, making com- 
parisons for the forty-eight states on ten different items, 
is found in Ayres's bulletin on the state school systems. ^ 

1 Page 32 




I a a o 

rt _ fe ^ o 

.2 .2 f^ a oj 

. .~ fl tn o) , 

o ^ c3 fl, a 

'^^^ ^ 

o c3 ,, ej 1^ 
•r -Q ti 

^ S 3 •-' 



g 73 
5 O fl 



<u 2 >> >> 
M g -a .1i 5 

O) " <D 4) 0) 

^■5 S 2 != 

j3 >^ <» S * 

-Q a ^ 02 

S o =* O 73 

■if ^ 03 QJ 

c .s « 3 
^ ^ 2 d « 

^ tn « a; Its 
^ a 

-w |_i IE S< =*-! 

•ft « Oi « ° 

a fl fe C! 2 

-►^ ^ « fa 

ro , ^ 4^ 

-fj t; ■-' ■ vi a> 

C O .ti T) ^> 

2^ ^ S & ra 
ft a> a >< "" 

57 , X « 4) 

t-i ^ o _fl 

a> '^ tc - ^^ 

c3 S 9 O O 

;i (- -u cc j3 
~ 3 tc o -t^ 

05 -^ W "^ -g 

«« a:S o -c 

0) 8:3:2 ^ 
-^ fl > S 

IK ir: ^j o 

■+^ •« -rH M 



o S 

OQ d 



245 



246 School Statistics and Publicity 

This is made up by starting at the top with the best state, Washing- 
ton, and going down to the lowest, Alabama. Each state is repre- 
sented by a separate bar, and each bar is divided into ten parts. 
Each of these ten parts represents an item on which the state is graded. 
For each item, the states in the lowest quarter have that part black ; 
the states in the next quarter have dark shading; the states in the 
next to the highest quarter have light shading ; and the states in the 
highest quarter have white. (See Figure 39.) 

This form of chart is excellent for graphing the numerous items of a 
summary table. The use of the different forms of shading uniformly 
through the table enables the reader quickly to locate any state's 
rank on any item. The use of the darker shades for the worst ranks 
on any item is also a good device, for it would have a tendency to 
sting the p^ide of the average citizen. Note also that an idea of the 
general standing of any state on the ten items can be gotten quickly. 
For example, Washington appears with practically white spaces and 
so is manifestly very high on the whole. Alabama has a black space 
on every item and so is very low. These things can be seen at a glance. 

Such a graph would be of use in comparing the per- 
formances of several pupils, of several teachers, of several 
classes, of several rooms, or of several schools where each 
individual case had been ranked on a number of qualities 
or achievements. 

For example, the showing on standard tests for several eighth 
grades might be made thus, using a horizontal bar for each grade, and 
a vertical column for each test used. 

The Boston Survey has a graph to show how Boston's 
expenditures compare with those of the average city, 
using component parts of a bar graph that is a square. 
It is reproduced here as Figure 40, but experiments show 
that while it is not readily understood at first by the 
average school man, it is exceedingly effective once it is 
grasped. 

Sometimes it is desirable to compare two distributions 
that have been grouped by similar steps, for the typical 



Graphic Presentations of School Statistics 247 



amount of the same quality in each step. In this case, 
a set of bar graphs running to the right may be used for 
one distribution, abutting on a similar set of bars running 

HIGH SCHOOL EDUCATION. PAYS 
YEARLY INCOME 



HIGH SCHOOL 
TRAINING 



NO K S. 

TRAINING 




N HIGH SCHOOL 



IN HIGH SCHOOL 



9 500 



1000 



I. 150 



^Z337 



H SCHOOL TRAINED BOYS-WAGES -53.50 PER DAY 

NO H. SCHOOL TRAINING -WAGES^ 1.50 PER DAY 

Fig. 41. — Example of Right and Left Device for Comparing Distributions 

with Bar Graphs. 

It shows average yearly income of high school graduates as compared with that of 
persons not having high school training. (From one of the 1917 folders of the Agri- 
cultural Extension Division of the International Harvester Company.) 

to the left for the other distribution. It is, however, 
difficult to compare magnitudes on the right with those 
on the left. 



248 



School Statistics and Publicity 



An example is the graph used by the Bureau of Edu- 
cation, shown in Figure 41. 

From this, it is only a short step to the graph which 
exhibits the facts about several qualities in a city or 

DECREASES INCREASES 



70 


6.0 


5.0 


• 4.0 


3.0 


2.0 


1.0 




1.0 


I II III 1 - 




' I^^Grade 




































— 


3rd 


















4+h 




































-,th 
















gth 
















oth 














■ 




- 10^^ 



















„tK 
















^^ 


,2th 



. Fig. 42. — Example of Right and Left Device for Comparing Distributions 

with Bar Graphs. 

It shows changes in distribution of enrollments by grades in Baltimore between 1899 
and 1909. (From Baltimore Survey, page 98.) 



school, by showing increases on bars extending to the 
right of a vertical line, and decreases by bars extending 
to the left of it. 

A good example is Figure 42 from the Baltimore Survey, showing 
tha change in enrollments in the various grades by percentages. The 



Graphic Presentations of School Statistics 249 

decrease in the first grade was due to special efforts to move up re- 
tarded children. 

This form is preferable to that in Figure 41 because the bars to 
be compared are closer to each other. 

The " Monument " Graph. Areas are not at all easy 
to compare. Such comparisons would be far more 
effective if made by bar graphs lined up at one end so 
that the comparison would merely be a matter of com- 
paring the lengths of the various bars. 

For example, take the "monument" graph which is frequently 
found in school reports. A good illustration is found in one repre- 
senting the number of pupils enrolled in each grade in the Alabama 
Survey of Three Counties, shown in Figure 43. 

Here each stone represents the enrollment for one grade, beginning 
with the first grade for 5423 pupils and topping off with a little stone 
for the 60 pupils in the last year of the high school. 

The errors in using this sort of graph are as follows: 
The areas of the " stones " may be taken into account 
for the relationship by the reader, when it really is shown 
by their lengths only. As there is no common point 
from which to measure either the length or area of the 
stones, no adequate idea of relative sizes can be obtained 
from such a graph. The effect of this form of comparison 
is to make the difference between the larger and the 
smaller numbers seem less than it is. This may be 
shown by rearranging the data in the graph in a regular 
bar graph lined up at the left. (See Figure 44.) Notice 
how much accentuated the differences appear. ' It may 
be contended that the monument graph is only one form 
of the graph given in Figures 41 and 42. That is, it 
really has two halves formed by an imaginary line down 
the center, either half of which gives a simple bar graph 
effect. But if that is the case, why not use the simple 



250 School Statistics and Publicity 



bar graph alone in the first place? When children are 
lined up in '' stair steps " to get their varying heights, 



IV H.S. 60 
JHH.S. I\9 
n H.S. 198 
I H.S. 455 



7^^(5r.Q\b 



^^ 6KI485 



b^^Qtr. 1800 



4^^6^2080 



3^Qr. EI32 



Z^Qr 2687 



|9* Gr. 54 Z3 



Fig. 43. — Monument Graph Showing Number of Pupils Enrolled in 

Each Grade. 
(From An Educational Survey of Three Counties in Alabama, page 63.) 

they are all placed on the floor. No one would ever think 
of putting their waist lines on the same level and then 
of taking account only of variations above the waist. 







ET H.S. 



60 



4ttj qrade 20bO 



1 



\ 1st grade M23" 



Fig. 44. — Graph Showing Apparent Size of Certain "Stones" from the 
Monument Graph Lined up at One Side. 

Comparisons with Circle Graphs. Errors frequently 
arise in making comparisons with circles. It makes all 
the difference whether the comparison is made through 
the diameters or through their areas. The ordinary 
reader tends to make the comparison on the diameter 
basis. If the circles are drawn on the area basis, how- 
ever, it is apparent that the comparison will not be so 
striking as the maker of the graph intended. If, on the 



Graphic Presentations of School Statistics 251 



other hand, circles are drawn on the diameter basis, some 
readers will tend to overestimate the facts. 

The graph used on page 91 of the Cleveland Survey, Summary 
Volume, is a good example of the difficulty in using circles on the area 




Under age and 
rapid progress 



© 



Normal age and 
rapid progress 



© 

Over age and 
rapid progress 






Under age a\nd 
normal progress 



Normal age and 
norma\ progress 



Over age and 
normal progress 



® 




Under age and 
slow progress 



Normal age and 
slow progress 





Over age and 
slow progress 



Fig. 45. — Comparison of Circles by Areas Using the Percentage of 
Children in Each Age and Progress Group in Elementary Schools of 
Cleveland at Close of Year 1914-15. 

(From Cleveland Survey, Summary Volume, page 91. By permission of the Siirvey 
Committee of the Cleveland Foundation.) 

basis. It is reproduced in Figure 45. Here nine circles are employed 
to show the percentages of children in the under-age, normal-age, and 
over-age subdivisions of the rapid, normal, and slow groups. This 



252 



School Statistics and Publicity 



requires nine circles and it is almost impossible to compare them ac- 
curately. One circle is marked 6 and another 30, but the former 
appears to be about one-fourth of the latter, or larger than in reality. 




A Larger Proportion of Children are Going to Nigh School. 

Fig. 46. — Concentric Circle Graph to Show Relative Increase in High 

School Enrollment. 
(From the 1915-16 Review of the Bockford, Illinois, Schools, page 107.) 



The use of concentric circles as a means of comparison 
is even worse than that of the circles apart from each 
other. 



Graphic Presentations of School Statistics 253 

Figure 46 is a concentric circle graph showing the enrollment of 
the Rockford schools for the years 1895, 1900, 1905, 1910, and 
1915, with a comparison of the high school enrollment with the total 
enrollment. This graph is worth little for the public, because it is 
hard to understand, and because it is almost impossible to get a cor- 
rect notion of the areas of the different circles. One tends to look 
only at the rings and not at the circles, and the whole effect is some- 
thing like the advertisements showing the various layers in an auto- 
mobile tire. 

The bar graph would be very much better for this 
comparison, as in Figure 47. 

Comparisons made by using the areas of segments of 
the same circle are questionable, because people have not 

Hig h School Grades 

1895 - 5141 

to 
1915 - e932 



Fig. 47. — Component Bar Graph Comparison of Data Shown in Figure 46, 

been trained to estimate the areas of segments. Even 
as simple a graph as " How Portland Spends Its Dollar " 
(see Figure 48) is hard to interpret correctly. But 
when the graph becomes as complicated as the one 
in Figure 49, it is probably useless for the average 
reader. 

It is even more difficult to compare sectors in different 
circles than in the same circle. 

For example, take the graph on the percentage of home-trained and 
non-trained teachers and principals in the Cleveland Survey, shown in 
Figure 50. The sectors here serve but little to emphasize the different 
percentages given. In addition, the white labels on the black parts 
probably cut down the apparent size of the black parts very mate- 
rially. 



254 



School Statistics and Publicity 



Triangle Graphs. A few surveys make comparison 
by the heights or areas of overlapping isosceles trianglet 
having equal bases, as in Figures 51 and 52. 



Interest 

207* 



Libra K[g5 1.01 



,1/ ^-^- 



.0'' 



A 



.^ 



<J^' 



/qa 



00 



Education 
30.7i 









% 



^<>?/> 






'Sf^ 



fo. 



\ 



■op_. 



Fig. 48. — Example of Difficulty of Comparing Component Parts in a 
Circle Graph When the Angles Are Not Clearly Shown. 

This figure gives an itemized statement of "How Portland Spends Its Dollar." 
(From Portland Survey, page 84.) 



Since two triangles with equal bases are to each other as 
their altitudes, the comparison is perfectly accurate from 
either a height or area standpoint. But as heights alone 
are really wanted for determining the areas, plain bars 




1/ 


y0(. 


\ 


\ 


^ 


X 


/ 


1 

(0 




\/\/X 


^ 


\ 


^ 


> 


^(O 


CM 




\/V^ 


I ^ 


V 


\ 


/ 


CO 


W 




V/\ 


\ 


\ 


\ 


/5 


1 


(U 




\\ 


\ 


\ 


y 


r 


(0 


o 

c 




lU 




o 


1 




"B* 

D 


c 




^ ^ 


lO 


IS 


03 


a> 


u. 




0> 


•5^ 


2 
o 


(0 

o 
o 


o 


5 




X.* 


52 


CO 


tJ 


CO 






Q> 


D 


3 




cd 









03 (1| 



"ft § 



^ <0 



CO 



a 



^1 



OS 

6 

M 



"Diagram III. Surface of circle represents total per capita expenditure in the 
average city. Sectors are proportional to amount spent for each of the twelve main 
purposes for which funds are expended. Shaded portion represents expenditure in 
Bridgeport. Under each heading the first figure gives in dollars and cents the 
amount spent per child per year in the average city and the second figure the corre- 
sponding amount for Bridgeport." 



255 



256 



School Statistics and Publicity 




Fig. 50. — Graph Showing 
DiflSculty in Comparing Sec- 
tors from Different Circles. 

(From Cleveland Survey, Sum- 
mary Volume, page 107. By per- 
mission of the Survey Committee 
of the Cleveland Foundation.) 



would show the relative lengths 
of the altitudes with much less 
work. Besides, since the aver- 
age reader has had no experience 
in comparing heights or areas of 
triangles, about the only justifi- 
cation for them is the one matter 
of adding variety. 

Comparisons with Cartoon 
Effects. Another bad use of 
comparison through areas is 
sometimes found in the employ- 
ment of cartoons in which the 
data are represented by the areas 
of persons or objects. For ex- 
ample, take Figure 53. The 
trouble with such a chart is 
that the area grows much faster 
than the height, so that the 
expenditure for 1914-15, instead 
of appearing only about 35 per 
cent greater than for 1910-11 as 
it should, really appears to be 
several hundred per cent greater. 
In this particular chart, the hori- 
zontal lines at the back help to 
reduce the exaggeration by em- 
phasizing the height factor. 

Even when the figures are 
given with the chart, as in this 
instance, the visual inaccuracy 
is serious enough to cause a dis- 
trust of the whole thing. How- 



Graphic Presentations of School Statistics 257 



ever, the cartoon effect can be secured in a graph that, 
by use of units or separate figures, allows no chance for 
error in making comparisons . 



)( 



A very effective chart of this kind is the one from the Des Moines 
^Report for 1914-15.^ It is too large to be shown effectively in this 
book, so will be described in words only. It aims to show in a car- 
toon the relative numbers of retarded children, normal children, and 



SALARIES OP 

PRIMCiPALS 




WAQES OF 
JANITORS 




Fig. 51. — Comparison by Triangles 
between the Salaries of Principals in 
Springfield and the Average for Ten 
Other Cities in 1911-12. 

Shaded triangle represents average annual 
per capita expense for principals' salaries for 
each child in average attendance in the day 
schools of Springfield, and triangle in outline 
represents corresponding expenditures for 
the average of ten other cities. (From 
Springfield Survey, page 98, by permission of 
the Russell Sage Foundation.) 



Fig. 52. — Comparison by 
Triangles of the Cost of Janitor 
Service in the Average City 
and in Bridgeport. 

Triangle in outline represents 
portion of each thousand dollars 
spent for janitors' wages in the 
average city ; shaded triangle repre- 
sents the amount spent in Bridge- 
port. (From Bridgeport Survey, 
page 27.) 



accelerated children, boys and girls separately. To the left, 27 boys 
and 20 girls are represented as climbing a hill, book in hand and read- 
ing. In the middle, 18 boys and 23 girls are walking along on a level 
with books tucked away under their arms. At the right, 5 boys and 
7 girls are going down hill with no books at all. This is similar to 
the Alabama illustration in that it gives relative proportions, but note 
how much better this relationship is shown by separate children than 
it would be by a few children varying in size. The graph would have 
been a little more effective if each group of children had been in one 

1 Page 99 



258 School Statistics and Publicity 

line so that the length of the line might also have entered into the 
comparison. This could have been easily managed by adding a few 
more lines to indicate a portion of a hill for each child not on the level. 

See how the child in the grades is growing 

1910-11 1911-12 1912-13 1913-14 1914-15 



t=rli 



$23^ $25^ $27^ 




Fig. 53. — Example of the Difficulty in Making Comparisons with Cartoon 

Effects. 

This shows what is spent in Louisville on each child in the grades. The comparison 
is really made by heights only, but the reader tends to take it by areas. (From 1914-15 
Louisville Report, page 35.) 

This would take no more room on the whole, and it would not seriously 
weaken the effect, since the hill conveys practically the same idea as 
the use of the books. 

Another example of a cartoon applying a good idea but in a very 
unreal way, and also using circles for comparison, is given in Figure 54. 



FOR EVERY DOLLAR THAT 

The average Bridofeporl 

cit y sp ends spends 

A 




BOARD OF EDUCATION OFFICE ^^^^H ^1 ^^ 22t 



Fig. 54. — Cartoon Graph Using a Sector of a Circle to Represent Part 

of a Dollar. 
(From Bridgeport Survey, page 22.) 

But the cartoon effect could have been kept and a much more accurate 
comparison made by using cents in a bar effect, thus : 

Average city $1.00 to 100 

Bridgeport .22 . . . to 22 



Graphic Presentations of School Statistics 259 



Ayres uses such a device to show the cost of schooling per child per 
day in the various states, in his bulletin on comparing the school 
systems of forty-eight states.^ 

When the cartoon effect is very necessary to show 
relative parts of a dollar, and accuracy is not essential, a 
cartoon similar to Figure 55 may be advisable. 

ADVENTURES OF MR. TAXPAYER. 



With a municipal budget and without. 





How one city now How a city 

SUPPOSES KNOWS 

the money is spent. where the money ^oes. 

Fig. 55. — Cartoon Effect to Show Parts of a Dollar. 
(From Newburgh Survey, page 93, by permission of the Russell Sage Foundation.) 

Figure 56 is a cartoon effect to show the lack of suffi- 
cient playground space. 

Figure 57 is a skillful use of a bar graph effect. 

Other good examples of cartoon effects may be found in : 

a. The Ohio Survey, page 65. 

Here there is a line of twelve teachers for each class of school, begin- 
ning with one-room township schools and going on up to high schooL 
Teachers without professional training are in black ; those with one 
or more terms in summer schools are in gray ; those with one or 

1 Page 18 



260 



School Statistics and Publicity 



more years in a professional school in white. A high degree of ac- 
curacy is attained by using half a figure to represent one twenty- 
fourth. Thus seven twenty-fourths of the twelve teachers in one 
line are represented by three women clothed in black and another 
with a black skirt and gray waist. 

LAWN vs. PLAYGROUND 

William Street 



How One 
Newbury 
School Saves 
the Grass at 
the Expense 
of the 
Children 




Fig. 56. — Cartoon Effect to Show Lack of Playground Space. 
(From Newburgh Survey, page 63, by permission Of the Russell Sage Foundation.) 



b. Ayres's A Comparative Study of the Public School Systems in the 
Forty-eight States, page 6. 

The value of school property in different states is represented by in- 
dividual dollar marks, a line for each state, giving a bar effect ; thus : 



Florida 
Kentucky 
Arkansas 
Mississippi 



IK (ft (J m (J (J* (J (P ^ (p (P <P ^ (P <P <p 

XO ip tp Jp fp ip ip ip ip ip ip tp ip tj) *f) «p 

"1 O (ft (ft <ft (ft (J (ft (ft (ft (P (P (P (J (P 

XO tptptptptptptpytptptptptp 

4 $$$$ 



Graphic Presentations of School Statistics 261 

c. The 1912 Newton, Massachusetts, Report, page 86. 

The enrollment of girls is represented proportionately for the dif- 
ferent kinds of schools. The lower grades are shown with smaller 
children and the high school with large ones. But the true relation- 



15 THE HIGH TAX CRY JUSTIFIED ? 



Totol Tax per Ccipifa ,\9\\ 



New Rochelle 




Hiag(^rQ Fa] Is Hi 


■■■■■■ 


Average 10 Ci\ie5 b| 


■■■■■ 19. 


Hiti^slon HH 


HHHl Id./ 


Pdu^hkeepsie HI 


HHHI 


Jamestown HI 


WKKM 13.56 


Hewbur^h |^| 


HHI 13.6 a 


Water town HI 


HHI 


AubuK/i HI 


WKM 12.48 


Amsferdam HI 


I 7.02 



25.10 



Fig. 57. — Illustration of Use of a Bar Graph for Publicity Purposes. 

Note that the figvires come at the right ends of the bars, but are much lighter in 
effect and so do not make the bars appear longer. (From the Newburgh Survey, page 
97, by permission of the Russell Sage Foundation.) 

ships are shown by the numbers of figures in the various groups. 
This gets all the value of the cartoon effect and avoids any trouble 
about comparisons through areas. 

Time Charts. A special device for comparing the use 
of something for several purposes, or the time put in by 
different persons, is the time chart. This is only one 



262 



School Statistics and Publicity 



form of a distribution table. The time element appears 
on one scale and the items on the other. The data to 
be entered are put in the rectangles determined by the 
two scales. These areas may be shaded, or they may 
simply be filled with words. Such a chart may be very 
profitably used to show how teachers spend their time, 
and it thus becomes simply the regulation school program ; 
or it may be used to show how much of the time various 
rooms are occupied ; or it may be utilized to show how 
much of the time the school plant or grounds are used 
during a year, etc. Table 42 is an example of this; 
the rooms could be listed down the side and the hours 
at the top if preferred. 

Table 42. Time Chart by Rooms and Classes 



Hours 


Room I 


Room II 


Room III 


Room I\ 


Room V 


8 


Math. 1 
Math. 1 


Eng. 4 


Hist. 2 


Sci. 3 


For. Lang. 1 


9 


Eng. 1 




10 


Math. 2 


Eng. 2 


Hist. 4 


Sci. 1 


For, Lang. 2 


11 








1 




Eng. 1 
Eng. 3 


Hist. 3 






2 


Math. 3 


Sci. 2 




3 


Math. 4 




Hist. 1 




For. Lang. 3 



Comparisons with " Curves." It is probably unwise 
to use curves often in presenting statistical material to 
the public. It is well for the superintendent to know how 
to employ curves of various kinds, for they are helpful 
in his own analysis of his data. Such uses have been 
shown on pages 166 ff. Engineers, draftsmen, economists, 
social workers, and other statisticians, of course, use 



Graphic Presentations of School Statistics 263 



them. They are to be seen in a few advertisements. 
But they probably have little significance for the general 
public as yet. 

At present, curves can probably be used most effectively 
for the public when the aim is to make comparisons be- 
tween the changes in the 

19 03-10. 1910-11. 1911- \Z 
$95 



TECH. HIGH 



NEWTON HIGH 



same classes of items on 
different dates. For this 
the same vertical scale 
may be used on either side 
of the diagram, with the 
years going out from left 
to right. 

A good example is found in 
the chart showing the decreas- 
ing cost per pupil for various 
groups, given in Figure 58. 
The chart is very fine, except 
that the omission of the zero 
line exaggerates the decrease 
altogether too much. 

A still better example of the 
use of curves for this purpose is 
Figure 59. 

Another example is a form 
showing the amount of money 
schools get as compared with 
what the city as a whole 
spends. 

Thus Professor Cubberley gives two graphs to show such facts for 
two cities. (See Figure 60.) 

Curve Effects with Bar Graphs. At present, instead of 
presenting curves to the public, it is probably better to 
use bar graphs arranged to secure the same general effect. 




ALL PUPILS 



- SHADES 



KINDERGARTEN 



Fig. 58. — Use of Curves to Compare 
Changes in the Same Classes of Items 
on Different Dates. 

The figure shows decreasing cost per pupil 
in various schools for the years 1909-12. 
Note omission of zero line. (From 1912 
Newton, Mass., School Report, page 35.) 



$65 
60 
55 
50 
45 
40 

35 
30 
25 
20 
15 
10 
5 



1907 '08 '09 '10 Ml '12 '13 '14 '15 

Fig. 59. — Use of Cvirves to Compare Changes in the Same Classes of 
Items on Different Dates. 

It show3 cost per pupil in grades and high school, 1907-15, Rockford, Illinois. Note 
zero line is given here. (From 1915 School Report, Rockford, Illinois, page 122.) 

264 

















/ 




C05 


r Per Pu| 


)il h 


igh ; 


ichoc 


^. 


/ 








y 














/ 


^^ 


/ 














/ 


















r 




































Cos 


,t ?ey 


Pup 


i! Ele 


ment 


3ry C 


Irade,! 


■■)y 








y 








f 


y 






— ■ ■ 


y^ 
































Cost 


Per P 


upil f 


or Ge 


neral 


Expe, 


ise.(Gr 


ades9 


H.S.) 




■ 

















Graphic Presentations of School Statistics 265 

Where two items are represented upon each of a number 
of bars, and the part representing one of these items is 
colored or shaded, the effect of two curves is obtained. 
The graph as it stands satisfies the untrained reader, 
while the trained reader can easily read off the curves. 
The diagram showing the retarded children in the grades 




Total City Tox Rate- 



s' fr5 

00 ^ 



^ CO t. 

< o :^ 



^^^^TSToTTorSXh^gFI^^^^^ 



Schenectady, fsf.Y 



§ § o ^ w CO 3: 
^ o^ o> ^ ijj 5^ O) 

San franc \SCO,CQi 



Fig. 60. — Example of Using Curves to Show Changes on Different 
Dates in Two Items for Two Cities. 

The two charts show the competition for city funds. (From Cubberley's Public 
School Administration, page 414, by permission of Houghton Mifflin Company.) 



of Memphis, Tennessee, public schools affords a fine 
illustration. (See Figure 22, page 166.) 

This same curve effect can be obtained by a trained man 
from the component part bar graphs. Obviously, if only 
two component parts are shown, the curve is read one 
way for one set, and another for the other, that is, looked 
at from the two sides separately, as in Figure 61. If 



266 



School Statistics and Publicity 



there are more than two component parts, the curves 
show only for the two end parts, as in Figure 62. 



c 




Fig. 61. — Curve Effect on Bars with Two Component Parts. 

The curve effect is also noticeable in a graph where 
various items in one group are compared with similar 
items in other groups. 

A good example is the standing in the four fundamental operations 
in the Courtis arithmetic tests for different grades. Letting A stand 
for Addition, S for Subtraction, M for Multiplication, and D for 




Fig. 62. — Curve Effect on Bars with Three Component Parts. 

Division, the results might be shown as in Figure 63. A line connect- 
ing the ends of all the bars of the same kind will give a curve. See 
also Figure 38. 

Standards for Drawing Curves. If, after all the pre- 
ceding, it is still felt desirable to present school statistics 
to the public with curves, the curves should be drawn 
properly. Accordingly, the suggestions of the Joint 



Graphic Presentations of School Statistics 267 



Committee on Standards for Graphic Presentation ^ 
should be followed. The words alone are given here, but 



No. 

16 
14 • 
12 • 

10 - 
8 • 
6 ■ 

A ■ 



probs. 

GRADE E" 



VI 





q\c. 



A 5 M D 



A S M D 



A S 



M 

Fig. 63. — Bar Graph Device with Curve Effect for Comparing Several 

Groups on Several Items. 

Each bar represents the standing in the Courtis Tests. A means addition, S, 
subtraction, etc. 

the full report contains graphs which make the text 
much clearer: 

1. The general arrangement of a diagram should proceed from 
left to right. 

2. Where possible, represent quantities by linear magnitudes, as 
areas or volumes are more likely to be misinterpreted. 

3. For a curve, the vertical scale, whenever practicable, should be 
so selected that the zero line will appear on the diagram. 

4. If the zero line of the vertical scale will not normally appear on 
the curve diagram, the zero line should be shown by the use of a hori- 
zontal break in the diagram. 

5. The zero lines of the scales for a curve should be sharply dis- 
tinguished from the other coordinate lines. 

1 Copies may be obtained from the American Society of Mechani- 
cal Engineers, 29 West 39th Street, New York, price 10 cents, discount 
in quantities. 



268 School Statistics and Publicity 

6. For curves having a scale representing percentages, it is usually 
desirable to emphasize in some distinctive way the 100 per cent line 
or other line used as a basis of comparison. 

7. When the scale of a diagram refers to dates, and the period rep- 
resented is not a complete unit, it is better not to emphasize the first 
and last ordinates, since such a diagram does not represent the begin- 
ning or end of time. 

8. When curves are drawn on logarithmic coordinates, the limiting 
lines of the diagram should each be at some power of ten on the log- 
arithmic scales. 

9. It is advisable not to show any more coordinate lines than 
necessary to guide the eye in reading the diagram. 

10. The curve lines of a diagram should be sharply distinguished 
from the ruling. 

11. In curves representing a series of observations, it is advisable, 
whenever possible, to indicate clearly on the diagram all the points 
representing the separate observations. 

12. The horizontal scale for curves should usually read from left 
to right and the vertical scale from bottom to top. 

13. Figures for the scales of a diagram should be placed at the left 
and at the bottom or along the respective axes. 

14. It is often desirable to include in the diagram the numerical 
data or formulae represented. 

15. If numerical data are not included in the diagram, it is desirable 
to give the data in tabular form accompanying the diagram. 

16. All lettering and all figures on a diagram should be placed so 
as to be easily read from the base as the bottom, or from the right- 
hand edge of the diagram as the bottom. 

17. The title of a diagram should be made as clear and complete 
as possible. Sub-titles or descriptions should be added if necessary 
to insure clearness. 

3. Special Summarizing Graphs 

Sometimes it is desired to give a graphic summary of 
something that has been measured by relative position 
on a number of items. The way of showing this roughly 
by the bar graph was given on page 244, but this indicated 
only the quarter of the distribution in which the case fell 



Graphic Presentations of School Statistics 269 




3 



4- 



lO 



1 1 



; 7. 



!3 



1 + 



J5 



I 7 



I 8 



19 



20 



21 



7 



I O 




/ 4 



\5 



t6 



I 7 



I 8 



IS 



20 



21 




lO 



I I 



12 



15 



IT 



I 8 



19 



20 



21 



Expenditure Expenditure per Expenditure ^Qr 

per $1,000 of tax- child in average 

inhabitant able wealth daily attendance 

The shaded rectangles represent Boston, 

Fig. 64. — Graphic Device for Summarizing the Relative Position of a 
Given Case in a Number of Different Distributions. 

This figure shows the rank of Boston in a group of twenty-one cities in expenditiire 
for operation and maintenance of schools. (From Boston Report, 1916, page 158.) 



270 



School Statistics and Publicity 



Teacher City 

(Indicate sex) 



EFFICIENCY RECORD 

Grade taught. 



(or building) (or subject) 

Experience years. Salary per month. 

Highest academic training '. 

Extent of professional training 



Detailed Ratixg V.P 



II. 



III. 



IV. 



General appearance 

Health 

Voice 

Intellectual capacity 

Initiative and self-reliance . . . . 
Adaptability and resourcefulness . 

Accuracy 

Industry 

Enthusiasm and optimism . . . . 

Integrity and sincerity 

Self-control 

Promptness 

Tact 

Sense of justice 

Academic preparation 

Professional preparation 

Grasp of subject-matter 

Understanding of children . . . . 
Interest in the life of the school . 
Interest in the life of the community 
Ability to meet and interest patroni 
Interest in lives of pupils . . . . 
Co-operation and loj'alty . . . . 
Professional interest and growth . . 

Daily preparation 

Use of English 

Care of light, heat, and ventilation . 

Neatness of room 

Care of routine 

Discipline (governing skill) . . . . 
Definiteness and clearness of aim . 

Skill in habit formation 

Skill in stimulating thought 
Skill in teaching how to study . 

Skill in questioning 

Choice of subject-matter . . . . 
Organization of subject-matter . . 
Skill and care in assignment 
Skill in motivating work . . . . 
Attention to individual needs . 
Attention and response of the class . 
Growth of pupils in subject-matter . 
General development of pupils 
Stimulation of community . . . . 
Moral influence 



General Rating 



Poor 



Medium 



Good 



Ex. 



Recorded by Position Date 

Fig. 65. — Summarizing Graph to Show Efficiency Record of a Teacher, 
Used by School of Education, University of Chicago. 



Graphic Presentations of School Statistics 271 



on that quality. A refinement of this is found in Figure 
64. 

Another example is that used by the School of Education 
at the University of Chicago to sum up the rating of a 



Grade 

Arithmetic 

Addition 

Spe&d 

Accuracy 

Subtraction 

Speed 

Accuracy 

Multiplication 

Speed 

Accuracy 

Division 

Speed 

Accuracy 

Silent Reading 

Handwriting 

Speed 
Qualify 



Grade n m m sr la im: 2m. 

Fig. 66. — Graphic Device for Summarizing the Achievements of One 
Pupil or School in Several Fields as Related to Standards in Those Fields. 

(From Educational Tests and Measurements of Monroe, De Vo33 and Kelly, by 
permission of the authors and Houghton MiflBin Company.) 

teacher on several different qualities. Part of this graph is 
shown in Figure 65.^ 

It should be noted that a general heading will loom 
up on this graph in proportion to the number of sub- 
heads it has. Consequently, the way to make any general 
heading have weight will be to add sub-heads, without 
caring particularly how important these sub-heads are. 

1 The original blank has suitable main headings for each Roman numeral, 
printed horizontally, but it was not feasible to show them on a page of this 
size. 




272 School Statistics and Publicity 

There appears to be no way of overcoming this defect 
except by varying the widths of the horizontal spaces, 
which would probably complicate the device beyond the 
point of practical value. 

A third summarizing device, and one capable of wide 
adaptation, is shown in Figure 66. The same device 
may be used to show the standing of a school or city on 
a number of items, by placing the names of the standards 
at the top in place of the Roman numerals for the grades. 

Figure 24 on page 168, showing the surfaces of fre- 
quency one above the other, is a fourth summarizing 
graph that is of value to the public. 

4. Brinton's Rules for Graphic Presentation 

Brinton, in his standard book. Graphic Methods of 
Presenting Facts, has a very convenient set of rules 
followed by a set of check items. These were printed 
before the suggestions of the Committee on Standards 
for Graphic Presentation (of which he is chairman) were 
published (see pages 267-268), and some of the suggestions 
appear in both sets. But Mr. Brinton's original lists 
are not confined to suggestions for curves, as is the report 
of the committee. The practical school man will prefer 
to have all the suggestions given in one place, so they are 
here appended. 

Helps on Graphic Presentations 

(Selected from Brinton : Graphic Methods of Presenting Facts, 

pages 360-362) 

I. Rules for Graphic Presentation 

1. Avoid using areas, or volumes, when representing quanti- 
ties. Presentations read from only one dimension are the least 
likely to be misinterpreted. 



Graphic Presentations of School Statistics 273 

2. The general arrangement of a chart should proceed from left 
to right. 

3. Figures for the horizontal scale should always be placed at 
the bottom of a chart. If needed, a scale may be placed at the 
top also. 

4. Figures for the vertical scale should always be placed at 
the left of a chart. If needed, a scale may be placed at the right 
also. 

5. Whenever possible, include in the chart the numerical data 
from which the chart was made. 

6. If numerical data cannot be included in the chart, it is well 
to show the numerical data in tabular form accompanying the chart. 

7. All lettering and all figures on a chart should be placed so 
as to be read from the base or from the right-hand edge of the 
chart. 

8. A column of figures relating to dates should be arranged with 
the earliest date at the top. 

9. Separate columns of figures, with each column relating to a 
different date, should be arranged to show the column for the 
earliest date at the left. 

10. When charts are colored, the color green should be used to 
indicate features which are desirable or which are commended, 
and red for features which are undesirable or criticized adversely. 

11. For most charts and for all curves, the independent variable 
should be shown in the horizontal direction. 

12. As a general rule, the horizontal scale for curves should 
read from left to right and the vertical scale from bottom to top. 
(See "special.") 

13. For curves drawn on arithmetically ruled paper, the ver- 
tical scale whenever possible should be so selected that the zero 
line will be shown on the chart. 

14. The zero line of the vertical scale for a curve should be a 
much broader line than the average coordinate lines. 

15. If the zero line of the vertical scale cannot be shown at the 
bottom of a curve chart, the bottom line should be a slightly 
wavy line indicating that the field has been broken off and does 
not reach to zero. 

16. When the scale of a curve chart refers to percentages, the 
line at 100 per cent should be a broad line of the same width as a 
zero line. 



274 School Statistics and Publicity 

18. If the horizontal scale for a curve begins at zero, the verti- 
cal line at zero (usually the left-hand edge of the field) should be 
a broad line. 

19. When the horizontal scale expresses time, the lines at the 
left-hand and the right-hand edges of a curve chart should not 
be made heavy, since a chart cannot be made to include the be- 
ginning or the end of time. 

20. When curves are to be printed, do not show any more 
coordinate lines than are necessary for the data and to guide the 
eye. Lines one-fourth inch apart are sufficient to guide the eye. 

21. Make curves with much broader lines than the coordinate 
ruling, so that the curves may be clearly distinguished from the 
background. 

22. Whenever possible, have a vertical line of the coordinate 
ruling for each point plotted on a curve, so that the vertical lines 
may show the frequency of the data observations. 

23. If there are not too many curves drawn in one field, it is 
desirable to show at the top of the chart the figures representing 
the value of each point plotted in a curve. 

24. When figures are given at the top of a chart for each point 
in a curve, have the figures added if possible to show yearly totals 
or other totals which may be useful in reading. 

25. Make the title of a chart so complete and so clear that 
misinterpretation will be impossible. 

Special. 

In showing deviations from a central tendency, on the vertical 
scale, upwards is plus, and downwards, minus ; on the horizontal 
scale, to the right is plus, and to the left, minus. 

II. Checking List for Graphic Presentations. 

1. Are the data of a chart correct? 

2. Has the best method been used for showing the data? 

3. Are the proportions of the chart the best possible to show 
the data? 

4. When the chart is reduced in size, will the proportions be 
those best suited to the space in which it must be printed ? 

5. Are the proportions such that there will be sufficient space for 
the title of the chart when the chart has been reduced to final 
printing size? 



Graphic Presentations of School Statistics 275 

6. Are all scales in place? 

7. Have the scales been selected and placed in the best pos- 
sible manner? 

8. Are the points accurately plotted? 

9. Are the numerical figures for the data shown as a portion of 
the chart? 

10. Have the figures for the data been copied correctly? 

11. Can the figures for the data be added and the total shown? 

12. Are all the dates accurately shown? 

13. Is the zero of the vertical scale shown on the chart? 

14. Are all zero lines and the 100 per cent lines made broad 
enough ? 

15. Are all lines on the chart broad enough to stand the re- 
duction to the size used in printing? 

16. Does lettering appear large enough and black enough when 
seen under a reducing glass in the size which will be used for print- 
ing? 

17. Is all the lettering placed on the chart in the proper direc- 
tion for reading? 

18. Is cross-hatching well made with lines evenly spaced? 

22. Are dimension lines used wherever advantageous? 

23. Is a key or legend necessary? 

24. Does the key or legend correspond with the drawing? 

25. Is there a complete title, clear and concise? 

26. Is the drafting work of good quality? 

27. Have all pencil lines which might show in the engraving 
been erased? 

28. Is there any portion of the illustration which should be 
cropped off to save space? 

29. Are the instructions for the final size of the plate so given 
that the engraver cannot make a mistake? 

30. Is the chart in every way ready to mark "0. K. "? 

5. Presenting Statistical Data with Maps 

Sometimes it is necessary to impress the public with the 
way items vary in size or frequency in different geographic 
areas. Thus, a state superintendent may wish to show 



276 School Statistics and Publicity 

the legislature how high school facilities vary in different 
counties of the state; a city superintendent may desire 
to demonstrate how far pupils have to go to reach a 
school, or the crowded conditions necessitating a new 
building in a certain locality ; or a county superintendent 
may need to show just how his county ought to be dis- 
tricted for schools so that all children will be within a 
reasonable distance of a school. These things may be 
shown fairly well by variations in shading the maps for 
different localities, a device much used by the United 
States Bureau of Census and by geography, history, and 
economics texts. However, all such maps require keys for 
their interpretation, and it is difficult for any one, except 
a person very familiar with such work, to understand 
them quickly. 

A much better way is to represent every case or every 
certain number of cases (say ten) by a dot. Figure 67 
is a map used by George Peabody College for Teachers 
to show the communities from which students have 
come to the college during the years 1914 to 1918. This 
illustration shows admirably just how much territory is 
coming under the direct influence of the school. It does 
not give a correct notion of the number of students 
that have attended the college during these years, as 
some cities have contributed possibly a hundred or more 
each and yet such a city would be represented by only 
one dot. If a dot had been inserted for each student, 
some portions of the map would have been solid black. 
This could be remedied by using perpendicular wires 
with a bead for each student, as recommended by Brinton,i 
but it is very difficult to reproduce such a map by pho- 
tography or by a drawing. In showing the widespread 
1 Graphic Methods of Presenting Facts, p. 251 




^ 






O lU 



a 

% 
o 

r/i 

(-> 
o 



o 



h o d 

.2 -si 



0) to 

OS 

a '" 

2a 

TO H 

o o 

0) 

2 f=i 
;:; o 

.2^ 



CO 



a; □ 
-^ 2 

"in 

a s 

.S eS 



277 



278 



School Statistics and Publicity 



influence of the college, the map is more effective than 
the one from the Report of the General Education Board 




Fig. 68. — Device for Showing Distribution of Cases on a Map. 

Each dot represents a student from the different counties, attending the University 
of Georgia. Note the radiating lines to indicate that the university is exerting an 
influence upon all the state. 

for 1914, page 4. This is merely a map of the United 
States in outline, with the number of students in each 



Graphic Presentations of School Statistics 279 

state attending Vanderbilt University. It is thus only a 
map with a table on it, and not even in good tabular 
form, because there is no direct way to compare the 
numbers by having them in one, column, and especially 
running from low to high or vice versa. It is impossible 
for one to visualize the number of students attending 
Vanderbilt University, and it takes a little time to get a 
correct notion of the sections of the country coming under 
its direct influence. 

The University of Georgia uses a good map to show its 
attendance, heightening the effect by adding radiating 
lines to indicate its influence in the state. (See Figure 68.) 

The dot device on a map of the city was used by the 
superintendent in the Rockford schools ^ to show the 
residence of pre-tubercular pupils and also the residence 
of students in the evening schools. In the one case, he 
showed clearly that the pre-tubercular children were 
scattered widely through the city and that the problem 
of handling them was city- wide. He also showed that 
the attendance at evening school was not restricted to a 
small area of the city. In some of the Red Cross work 
huge state maps are shown with the number of tubercular 
soldiers for each county pasted in as so many paper doll sol- 
diers, upright in ranks. It is a very effective presentation. 

If the distances on a map are to be measured in some 
time unit, a map similar to that in Figure 69 is useful. 

The accredited schools of a state may be shown on an 
outline map, using different-colored pins or tacks for the 
different classes of schools. In the division of rural 
education in the state department of education for 
Missouri, there is an immense map of the state painted 
on the wall. On this, each approved rural school is shown 
1 Rockford Review for 1915-16, pp. 85, 104 



280 



School Statistics and Publicity 



by a small kodak picture of the building. This enables a 
visitor to get almost immediately an idea of where the good 
rural school work is being done, and closer examination 
will show the kinds of school buildings being put up. Of 




• Fig. 69. — Device for Showing Distance with a Time Element on a Map. 

Map iised by the Nashville Commercial Club to show how accessible the city is. 
Every city indicated is within twelve hours' travel of Nashville. 

course, such a map cannot be easily reproduced in printing. 
Still, something can be done with conventional drawings 
for each item in a class. Most readers of this, for example, 
probably recall the recent Y. M. C. A. propaganda with 



Graphic Presentations of School Statistics 281 

the sketch of the Western Battle Front, each building of 
the organization being represented by a tiny drawing of 
the right kind. 

III. HOW GRAPHS FOR THE PUBLIC DIFFER FROM THOSE 
FOR THE ADMINISTRATOR 

The difference has been referred to many times before, 
especially in discussion of the superior value for the 
public of bar graphs over curves and in Brinton's check 
list.i Let us now analyze it further, restating certain 
points for additional emphasis. 

In general, the public will view graphs in much the 
same way as they view any explanation or presentation. 
The ordinary man cannot quickly get from a rough copy 
of a chapter the meaning that an experienced writer can ; 
he cannot extract from a confused and verbose mass of 
evidence the essentials that a trained lawyer can; he 
cannot grasp so quickly, nor in such large numbers, 
many intricate graphic presentations that seem relatively 
simple to a trained school man. School^graphs for the 
public must be simple, with^jrelatively few" elements or 
lines, and very forcible. The trained reader can extract 
the significant things from most graphs, however poorly 
constructed ; the average man cannot. Let tfe^now take 
up some of the most significant aids for making graphs 
clear to the public. 

1. The title should give all the significant points to he 
found in the graph, so that the graph would he quite in- 
telligihle apart from the context where it is found. 

This implies that there should not be many different points found 
in one chart. If the chart can show only one thing, it is all the better. 
Some examples of good titles are : 

1 See pp. 274-5 



282 School Statistics and Publicity 

Standing of the children of Salt Lake City in the fundamentals 
of arithmetic, judged by the median score attained by each grade. 
{Salt Lake City Survey, p. 175.) 

Distribution of ages at which Salt Lake City children enter the 
first school grade. (Same, p. 201.) 

Columns represent number of pupils among each hundred begin- 
ners who remain in school at each grade from the first elementary 
to the fourth high. {Cleveland Survey, p. 88.) 

Percentage of elementary teachers, high school teachers, and 
elementary principals in Cleveland who are home trained and not 
home trained. {Cleveland Survey, p. 107.) 

How Portland spends its dollar. {Portland Survey, p. 84.) 

Figure 7, representing the percentage of children in several 
grades who make the given scores in composition. For ex- 
ample, 1.7 per cent of the fourth-grade children wrote com- 
positions scored at 0; 43.8 per cent of the fourth grade were 
scored at 1; etc. By following the median lines, the overlap- 
ping of ability from grade to grade is disclosed. {Butte Survey, 
p. 75.) 

In each of these instances, the title and the chart make a complete 
unit. The last one is especially noteworthy. It gives a very brief 
title, then expands this with a full but concise explanation of all points 
in the diagram that may cause confusion. 

2. A chart or graph too large to he seen without turning 
the head is apt to he a poor chart for the puhlic, no matter 
how simple it may he. 

This is shown by some charts and folded graphs in some of the 
surveys. It is very seldom that a chart or graph should occupy more 
than one page of the publication in which it appears. 

In reducing charts for publication, however, one must be careful 
that the reduction is not carried to the point of making the differences 
negligible or the lettering too small to be read easily. Advertisers 
long ago discovered that the public will not read advertisements in 
fine print, and school graphs are only a form of advertising the school 
work. Often graphs are reduced greatly to economize space. If 
this is pushed to the extent of making the graph hard to read, it is 
clearly advisable to omit some of the graphs altogether, and make the 
remainder large enough to be effective. 



Graphic Presentations of School Statistics 283 

3. The background of a chart should not he made any 
more prominent than necessary. 

Many charts are plotted on coordinate paper heavily and finely 
ruled, while the curves or bars are but a trifle heavier than the co- 
ordinate ruling. Such charts do not stand out clearly from their 
background. Only as many coordinate lines should appear on a 
chart as are necessary to guide the eye of the reader and to permit of 
easy reading of the curves. 

The difficulty may be rather easily avoided by drawing on the 
coordinate paper in very heavy lines with India ink all the lines and 
figures which it is desired to reproduce, including the coordinate 
ruling. The necessary lettering can be put in with a typewriter, using 
a practically new black ribbon. The proper exposure in making the 
cut will "take" all the desired lines and lettering, but not the others. 

4. Exaggerations should he avoided as much as possible. 

(a) Usually, a very forceful presentation may be had without any 
great sacrifice of accuracy. 

Of course, only complete scientific presentation can ever give 
the whole truth. However, if only a phase of a problem at a 
time is presented to the public (and often this seems necessary) , 
some exaggeration is inevitable. Here the problem is to choose 
between absolute accuracy and forcefulness of presentation. 

(6) It is, in general, dangerous to leave the zero line off a chart 
intended for the public, or even to send it out with the conventional 
wave line at the bottom. 

Many school men in making charts have found it convenient 
to leave the zero line off. Sometimes, when all the quantities 
used come high on the vertical scale, this is done to economize 
space. 

An excellent example of such a chart is Figure 70. The 
upper chart conveys the idea that salaries have increased greatly 
in Louisville during this period of five years. But the chart 
begins at about $475 instead of zero. The chart below is drawn 
in full. It is clearly seen from the complete chart that the salary 
increase, while noticeable, does not appear anything like so large 
as in the original and incorrectly drawn chart. 



284 



School Statistics and Publicity 



(c) Care should be taken that the scales chosen for the graph do 
not exaggerate things unduly. 

The novice in chart making will probably become confused by 
the ever changing ratios between the perpendicular and the hori- 
zontal scales. No definite rules can be laid down for guidance in 

^5 if was drawn 



^550 $600 -1650 itJOO 


19*0-11 1 








1 19/1-12 1 








1 I9IZ-I3 1 1 








[ igi3-i4 1 1 








1 19(4-15 1 1 1 













As 


t should 


hdve 


been 














IJO ~?|00 fl50 4200 ^250 ^300 4350 f400-#450 -?500 #550 ^600 ^650 ^700 


I9l0|-ll 




1 




1911 


-IZ 






















1912 


-13 






_ 
















i9(3 


-14 






















1914 


-15 




^ 


^ 


^ 

















Fig. 70. — Example of Danger of Leaving Zero Line Off a Chart. 

The top set of bars was intended to show a "comparison of salaries paid to elementary 
school teachers in Louisville for the years 1910-1915." The bottom set shows what the 
comparison really was. (Adapted from the Louisville Report, 1914-1915, page 18.) 

this matter. The only way to get facihty in adjusting these 
. ratios in the proper way is through the trial and error method. 
That a change in the ratio makes a great difference in the im- 
pression produced by the graph is shown in Figure 71 



Graphic Presentations of School Statistics 285 



The left-hand graph shows the results of the Kansas Silent 
Reading Tests in the Rockford schools. The right-hand graph 
shows the same data plotted with the horizontal scale increased 
while the vertical scale is decreased. It will be seen at once 

Scole Grade 

345 6 7 8 
20 



10 




5 



Scale Grade 

3 4 5 6 7 8 



15 



10 




Fig. 71. 



Example of Effect Produced by Changing Ratio of Horizontal to 
Vertical Scale on a Graph. 

Results of Kansas Silent Reading Tests. (Adapted from Review of Rockford, Illinois, 
Schools, 1915-1916.) 



286 School Statistics and Publicity 

that the difference in achievement between the grades does not 
appear so marked in the second illustration as in the first. 

(d) The superintendent must beware of graphs containing optical 
illusions. 

There is little danger of this in charts made small enough to 
publish, but there may be danger in large wall charts. Brinton 
calls attention to illusions caused by a row of perpendicular lines 
compared with a row of horizontal ones the same distance apart. ^ 






Fig. 72. — Cartoon Graph Representing Ratio of Lighting Space to Floor 

Space. 

(From Alabama Three-County Survey, page 92.) 

The lines in the first row appear shorter than they really are and 
spread farther apart ; those in the second seem to be longer than 
they really are. Another illusion is caused when a white square 
and a black one of exactly the same size are drawn adjacent. The 
white one seems larger. This might affect slightly the ratio of 
lighting space as shown by white windows against a dead black wall 
space, in some surveys. (See Figure 72.) 

5. Special effort should he made to introduce variety, 
novelty, and various striking features to attract attention 
to the statistical relations to he presented. 

Variety in graphs is as necessary to keep the attention as is variety 
anywhere else. No ordinary reader could stand it to wade through 
a report of any length which had a great many bar graphs of one 
pattern and no other illustrations. 

Sometimes pictures or devices may be used simply to catch the 
attention. The bulletin on illiteracy in Virginia, 2 for example, uses 
a picture of a rural school to get the reader's attention for the state- 

1 Graphic Methods of Presenting Facts, p. 358 

2 Illiteracy in Virginia, published by State Department of Public 
Instruction, p. 7 



Graphic Presentations of School Statistics 287 



THE PROFIT FROM TWO HERDS FOR ONE YEAR 



rr 









$95.73 






._.^j l Ml ' I I I I I 

'■"•' ' State Bank 



WHY THIS DIFFERENCE? 

Herd A 



IT WAS NOT THE SIZE OP HERD .11 COWS 

9 PURE Bred 

It was mot the breed % grades 

IT was mot the feed cost -♦sze.yo 

(silos and cood buildings on each farm) 

HERE IS THE ANSWER 

AVCRAGIE PRODUCTION OP 

BUTTER FAT I7I.I LBS. 

PER COW 

This Is A True Story As Told Dy 



Herd B 



II COWS 

t NATIVE 
10 GRADES 

$569.96 



386.9 LBS 
PER COW 




*^n 1 ■ I J ' ^ ■ * I t,IZI 



^ 



H0RAL:-IT would have taken 93 POOR cows TO MAKE 
THE PROFIT THE 11 GOOD COWS MADE; 

DOES IT PAY TO KEEP RECORDS? 



Fig. 73. — Example of Use of Special Devices to Attract Attention to the 

Statistics Involved. 

This figure compares the profit from two herds for one year and shows how many 
dairymen are wasting time and money on low-producing cows. "Why not get rid of 
your 'visitors'?" (From a pubUcation of the University of Wisconsin, Experiment 
Division, by permission.) 



288 



School Statistics and Publicity 




ment that only six of these children are beyond the first reader and 
none beyond the fifth reader. 

Or devices similar to those in Figure 73 may be used to attract 
attention to the statistics. 

Then there are special touches which no one can tell precisely how 
to go about acquiring. For example, in the Alabama three-county 

A ^ ^ ^ 

ftfl 

One outof every ten white men must ask another 
to mark his ballot for him. 

ttMMIII 

Foor outof every ten negroes must ask another 
to ^ign their names for them. 

Fig. 74. — Bar Graph with Cartoon Effect Showing Illiteracy in Alabama. 
(From the Survey of Three Counties in Alabama, page 19.) 

survey, illiteracy is shown among voters by having the literate voters 
face the reader and the illiterate ones turn their backs. (See Figure 74.) 
Probably this is about as forceful a showing of the shame of illiteracy 
as could be devised. 

IV. EXAMPLES OF GOOD GRAPHS ON SCHOOL STATISTICS 

FOR THE PUBLIC 

1. To Show Rise in School Costs 

In the Newton, Massachusetts, Report for 1912,^ the 
scale on tax for school maintenance per $1000 is shown 
on a thermometer device which is reproduced on page 102 

1 Page 113 



Graphic Presentations of School Statistics 289 

of this book. This gives the idea that costs for school 
maintenance should rise. The names of various cities 
with which Newton is compared appear at one side of the 
graph, with lines running from each name to the proper 
degree on the thermometer where the mercury should 
stand for that city. This graph shows very forcibly that 
the mercury must rise many degrees for Newton before 
it will equal the best record made by the other cities. 

For some cities, probably a cartoon utilizing the weigh- 
ing machine seen at fairs and carnivals would be equally 
effective. 1 Instead of pounds, tax leyies or per capita 
amounts of money could be shown on the scale of the 
upright, with the highest amount reached or desired 
at the top. Men representing the other cities could be 
standing around, evidently having struck the machine, 
and their records could be shown on a bulletin board in 
the background. Another man, representing the home 
city, could be shown as just getting ready to strike the 
machine to see what he can do, in the midst of words of 
encouragement or taunts from the other men. His old 
record might appear on the bulletin board. Underneath 
might be some such question as, " Can't he send it to the 
top? '' " Who is the best man? " '' How much will he 
beat his old record? " 

2. To Show Relative Investments in School Property 

In the Educational Survey of Three Counties of Alabama, 
the number of dollars invested for each child of school 
age by each state is given. ^ Each dollar is represented by 
a dollar mark. Thus, Massachusetts has $115 invested 
in school property for each child of school age, while 

1 Suggested by Mr. F. C. Lowry 2 Page 212 



290 School Statistics and Publicity 

Mississippi has only $4. The advantages of this graph 
are : The sjmibol aids in calhng attention to the graph ; 
the length of the row of dollar marks gives the effect of a 
bar ; the data are accurately represented, — there is no 
material exaggeration or anything in the device to mis- 
lead. 

3. To Show a Lack of Funds for Maintenance 

In the Survey of Three Counties of Alabama,^ there 
appear pictures of a schoolhouse and an automobile. 
From suitable figures, we learn that the initial cost of a 
cheap automobile is more than that of the average rural 
schoolhouse; and that the upkeep of the machine is 
more than that of the average rural school. This com- 
parison depends for its power on contrast, not on accuracy, 
for there is nothing particularly accurate about it. All 
the same, it is a very powerful device in shaming rural 
people into doing their duty by schools. If there is a 
single automobile in such a district, it represents more 
than the rural school expenses. This illustration will 
be of service chiefly in suggesting similar comparisons. 

The Columbus Dispatch some months ago had a very 
effective cartoon to represent the disparity in wages of 
women teachers and statehouse janitors in Ohio.^ It 
depicts a gruff old man with hands in his pockets, labeled 
" Old Man Ohio.'' Above him are two inserts. The 
left insert represents a pitiful woman teacher in her 
classroom, with the statement that the average salary 
for the public school teacher in Ohio is $54 a month. The 
right insert depicts a shuffling negro janitor in cap and 
overalls, bearing broom, mop, and bucket, with the 

1 Page 72 

2 Reproduced in American. School Board Journal, Jan., 1917, p. 33 



Graphic Presentations of School Statistics 291 

statement that Ohio pays the janitors in the statehouse 
$60. On either side of Old Man Ohio is a hand with index 
finger pointing at him to emphasize the title of the whole 
— " For Shame ! " 



4. To Show Length of School Term, Average Attend- 
ance, Etc. 

For this the Ayres bulletin on the forty-eight states has 
a good graph. Each day is represented by a small square, 
the whole representing a bar graph, with each bar two 
squares wide to make the bars shorter. The total length 
of the bar represents the average number of days in 
which schools were open in that state. Beginning at the 
left, a sufficient number of these little squares are shaded 



40 60 60 

49- NEW MEXICO, 






Fig. 75. — Graph for Showing the Relation of the Average Number of Daj^s' 
Attendance by Each Pupil to the Number of Days School Was Open. 

(From Dr. Ayres's Comparative Study of the Public School Systems in the Forty-eight 
States.) 

to represent the average number of days attended by each 
pupil enrolled in that state. The names of the states 
appear on the left from high to low, beginning with Rhode 
Island, which had her school open 193 days and kept 
each pupil in 148.8 days. The lowest is New Mexico, 
which had her schools open only 100 days and kept each 
pupil in only 66.4 days. This is represented in Figure 75. 
This chart shows clearly which states have the longest 
school terms and which are making the best use of what 
they have. It could be used for cities just as well as 
for states. 



292 School Statistics and Publicity 

Another way to show the attendance of different school 
systems is suggested by the graph on page 10 of the same 
bulletin. This shows the number of days of schooling 
each child of school age would get per year if he got his. 
share. Each day is represented by a small dot; the 
dots are clustered in groups of five, thereby giving a 
unit of the week as well as the day. The bar effect is 
obtained, and the graph has every advantage mentioned 
for those above. This graph could be used in any graph- 
ing of attendance, probably. The copy used by Ayres 
has the figures at the right end of the bars, which is bad, 
because it makes the bars appear lengthened unequally. 
This is corrected in the copy below. • 

48. New Mexico 46 :•::•::.::•::• : etc. 

5. To Show Per Capita Costs of Schooling 

A graph for this, similar to the last two described, is 
found on page 18 of the Ayres pamphlet. It sets forth 
the cost of one day's schooling for one child in each state 
in 1910. Each cent is represented by a black dot, and the 
bar effect is obtained. This dot or the cent mark or the 
dollar mark could be used in gi-aphing any data on costs, 
the unit being chosen so as to keep the bar short enough.^ 

South Carolina 7 :•::•::.::•::• : etc. 

6. To Show a Disgraceful State of Affairs in Certain 

Localities 

On page 160 of the Alabama three-county survey is 
shown a familiar map graph. This particular one shows 
the map of the United States with all states having com- 

^ See page 260 of this book 



Graphic Presentations of School Statistics 293 

pulsory education laws in white, and those not having 
such laws in black. As black is usually associated with 
shame and disgrace, the graph becomes a stinging accuser 
against the sections that are backward in this respect. 
The same idea, of course, has been used in religious maps. 
This use of black was referred to in the description of the 
chart on page 32 of the Ayres pamphlet. (See page 
244 of this book.) The idea is capable of wide use in 
cases where it is desirable to shame backward school 
systems into doing something better. The objection that 
it is difficult to show lettering on black areas is easily over- 
come by using white ink for lettering. 

7. To Show the Variability of Children in the Different 
Grades in Their Achievement in Some Standard Test or 

Similar Matter 

A discrete distribution or Bobbitt table may be shown 
with bar graphs as shown on page 103, with a curve as 
shown on page 104, or with, a scale as shown on pages 101 
and 102. 

For a continuous distribution, some form of the block 
graph is probably best. It is much used in the surveys of 
Butte, Salt Lake City, and other places. (See pages 112 
and 137 of this book.) 

8. To Show the Relation between the Rank of Children 
in Their Classes in the Elementary School and Their 

Probable Entrance into High School 

The pupils are divided into three classes, the upper, 
middle, and lower thirds. Each third is represented by a 
broad vertical bar, all bars the same length. The per- 



294 



School Statistics and Publicity 



centage of each bar representing those not going on should 
be colored black or shaded, the rest being left unshaded. 
The three bars should be placed side by side, with the 
low third on the left and the high third on the right. (See 
Figure 76.) 




Low Middle High 
third third third 



Fig. 76. — Graph Showing Percentage of Eighth Grade Pupils Entering 
High School from the Low Third, the Middle Third, and the High Third 
of their Classes, Cleveland. 

(From Cleveland Survey, Summary Volume, page 185, by permission.) 



9. To Compare Achievements of the Various Grades in 
Standard Tests with Similar Grades from Other Cities 

The best graph for this purpose is probably a modifi- 
cation of the bar given in Figure 63 on page 267. Each 
grade could be represented by the one kind of shading 
throughout. Each group could have the name of its 
city beneath, the scale could appear on each margin, and, 
if necessary, faint horizontal scale lines could be drawn 
clear across the drawing. 



Graphic Presentations of School Statistics 295 

10. To Show Ratio of Lighting to Floor Space 

In the Springfield Survey, sl graph is used for this in 
which the floor space is represented by a black square. ^ 
In the midst of this is a white square representing the 
lighting space. In the first diagram to the left appears 
the standard ratio ; the second one gives the average ratio 
for Springfield. The percentages appearing in the white 
squares should be written below. White squares always 
appear larger than black ones in such a graph, but this 
one is too small for the feature to affect it materially. 
The white square in the midst of the black gives a window- 
like effect and so helps to call attention to the diagram. 
This may be improved upon by making the whole the 
shape of a side wall, with the white the shape of windows 
in that wall. (See Figure 72, page 286.) 

The preceding illustrations comprise only a few of the 
best selections from school reports and surveys, in addition 
to those given before. It will be noticed that many of 
the problems the superintendent is sure to meet in 
graphing his data have not been mentioned. The bar 
graph, concerning which much has been said in the pre- 
vious pages, is capable of wide adaptation, as is also the 
Bobbitt table. One or the other of these two may be 
pressed into service upon almost any occasion. It is 
well, however, to introduce some of the special forms 
mentioned above, for variety's sake if nothing else. 
The efficient superintendent, of course, will always be on 
the lookout for improving the devices we now have for 
presenting statistical data to the public, or he may work 
out some entirely new methods. But any graph he 

1 Page 24 



296 School Statistics and Publicity 

devises for the public should be relatively simple, very 
clear, and, if possible, forceful. 

V. ECONOMIES IN MAKING SCHOOL GRAPHS FOR THE 

PUBLIC 

Large Cross-Section Paper. If large charts are to be 
made for display purposes, excellent results may be 
obtained from the use of good wax or grease crayons of 
assorted colors. The time required for getting accurate 
measurements may be greatly reduced by the use of 
large size cross-section paper, as the counting is then very 
easily done for the two scales or for locating any particular 
point. The paper for this should be heavy manila, 
light enough in color and sufficiently free from spots to 
make a good background for the colors, and rough enough 
to take the colors easily. Sheets 36 by 40 inches ruled 
faintly in one-inch squares give excellent results. 

These may be obtained from any large printing house equipped 
with ruling-pen machines, but are very expensive in small quantities. 
The University of Chicago Press carries them in stock at from three 
to five cents per sheet, depending upon the price of paper, transporta- 
tion extra ; or they may be obtained from the Peabody College Book 
Store on the same terms. 

Making Cartoons. In some cases, good results may 
be obtained by pasting pictures on charts, if the cartoon 
effect is desired. 

For example, the writer wished to reproduce in large form the 
automobile and schoolhouse graph from the Alabama three-county 
survey, referred to on page 290. He got a picture of an automobile 
from a large-sized advertisement in the Saturday Evening Post and a 
picture of a rural schoolhouse from the front cover of an American 
School Board Journal. By putting on the title and the figures, the 
chart was soon made. Students often employ this same device in 
getting up posters for school entertainments. 



Graphic Presentations of School Statistics 297 

Gummed Letters. Much time is saved and a beautiful 
chart may be made at little cost of time and money by the 
use of gummed paper letters and strips. These can be 
gotten in several colors and any height from four inches 
down. The superintendent can use letters from one to 
one and a half inches high, costing from about $1.50 to 
$2.00 per thousand. The letters may be spaced very 
easily and quickly if the chart is made on a large sheet 
of cross-section paper. 

These letters and strips may be obtained from the Tablet and 
Ticket Company of Chicago, which will send pictures of such charts. 
The strips are very serviceable for making bar graphs. 

Rubber Stamp Set. An advertising set such as is used 
by merchants for display cards can be used advantageously 
in making good charts. With a little practice, neat 
charts can be quickly made with such a stamp set, 
especially on the large cross-section sheets. 

A satisfactory set may be obtained from the Milton Bradley 
Company, 73 Fifth Avenue, New York City, or from Salisbury- 
Schulz Company, 157 West Randolph Street, Chicago, or at any of 
their various offices, for about $5, depending upon the price of 
rubber. 

Securing Clear Lines. In drawing charts for cuts, 
the chart and all essential cross-section lines should be 
reproduced. The error of reproducing the background 
of numerous cross-section lines may be avoided by tracing 
over in India ink the parts that should come out in the 
cut. The photographic process will reproduce this easily 
before the background with its fainter tones will '* take.'' 
The too prominent background is usually due to making 
the drawings with ordinary ink or faint typewriter letter- 



298 



School Statistics and Publicity 



ing. The latter may be easily traced in India ink and 
give good results. (See page 76.) 

Miscellaneous Aids. Numerous aids to making charts 
quickly with various kinds of cross-section paper, special 
scales, etc., may be obtained from the Educational 
Exhibition Company, Providence, Rhode Island, or the 
Tablet and Ticket Company of Chicago. 

" Perpetual " Attendance Graph Device. An exceed- 
ingly easily managed graph arrangement was observed 




Fig. 77. — "Perpetual" Attendance Graph Device. 

some years ago by the writer in the device used by 
Principal R. L. Dimmitt of the Ensley High School at 
Birmingham, Alabama, to compare the attendance 
record of the classes in the high school. (See Figure 77.) 



Graphic Presentations of School Statistics 299 

A large chart was made up on Bristol board, once for all, with the 
percentage scale running up on the sides. Each class was repre- 
sented by a paper ribbon that came through a slit on the base line. 
By pulling the ribbons up and down each month and fastening the 
ends with thumb tacks, the graph was quickly brought up to date. 
The omission of the zero line exaggerated differences, but such ex- 
aggeration was to some extent desired for emphasis. The only real 
drawback seems to be that one cannot compare the records of the 
classes by month with such a chart. The chart as operated, however, 
does not need to show comparisons. The emphasis on attendance 
is intended to keep up attendance all the time, and not to let children 
slack up one month because of a good record the preceding month. 
If, however, it is desirable to make comparisons, this can be easily 
done by preserving kodak pictures of the chart at various times. 
This cut is drawn from such a picture. 

Graphs without Cuts. If the charts are to be set up 
in type and not with cuts, the originals may be made 
either by hand or on the typewriter. For all such work 
the horizontal or vertical bar graph is especially useful. 
By the aid of conventional signs, different lengths of 
rules, etc., almost any chart containing a bar effect can 
be made so simple that it can be set up in any ordinary 
limitedly equipped printing office. The following are 
examples : 



XX 

xxxx 

j{)jj>;}>;j>Jt) xxxxxx 

*mmm . xxxxxxxxx 

XXXXXXX r^^^^-r^TTTTrTrrT^^^^^^^^^ XXXXXXXXXXXX 

0000 xxxxxxxxxxxxxxxx 

-^- ______-_-_-_-_-_------ xxxxxxxxxxxxxxxx 



Utilizing Students. In most of the work on graphs for 
the public, the superintendent need only furnish the idea 



300 



School Statistics and Publicity 



or design for the chart. The rest of the work can be 
done by various upper grades or high school classes as 
very profitable laboratory or practical exercises. The 
mathematics classes, especially those in algebra, and the 
drawing and art classes are the logical ones to call upon 
to take charge of this work. 

In Newton, Massachusetts, the boys in the high school printed 
the title pages of the annual school reports and probably made the 
graphs, although this latter is not so stated. In the World Book 
Company's reprint of MacAndrew's The Public and Its School, the 
drawings are all made by public school children. Children who could 
do such drawing could make most of the graphs advocated in this 
book. As it is, practically all the drawings in this book have been 



(^^^W^^Es^iti: 




rtSTS, BCTFt 5UR.VEY^ p. 93. [^fy\\ pupi I^ ICsteJ ) 

Fig. 78. — Showing Sketch and Rough Notes from Which a Pupil Drew 

Figure 13. 

copied or drawn from the author's suggestions and statistical data 
by the students of Mr. E. S. Maclin at the Atlanta Technological 
High School, as a demonstration. Figure 78 shows the sketch and 
notes furnished one of these pupils by the writer, from which Figure 
13 was drawn. Figure 79 shows the cartoon drawn by a student 
from notes given by Mr. Maclin. 



Graphic Presentations of School Statistics 301 



ONLY TOO TRUE 



cw>. I'm growing 

|50 FAST THAT ILL 

SOON be: in ThEl 

^TREET AND NO J 




Fig. 79. — Cartoon Drawn by an Atlanta High School Student. 

This cartoon represents the high school situation in Atlanta and was drawn by the 
student with only these suggestions from his drawing instructor, Mr. E. S. Maclin : 

Subject : The overcrowded conditions of the Atlanta High Schools. Represent a 
good-natured boy who has outgrown his clothes at every point, trousers splitting, 
feet running out of his shoes, shirt too small for him, etc. He is represented as 
saying to his father, the City Fathers, "Dad, I'm growing in spite of you. I'll 
soon be in the street and no place to go." His father is represented as being a 
rich man with Atlanta's per capita bonded indebtedness of about $22 per in- 
habitant. 

It will be noted that the suggestions were not wholly followed, but the cartoon as 
published in an Atlanta paper was sufficiently forceful. 



302 School Statistics and Publicity 



EXERCISE 



1 



Take the school report or school survey used in "the exercise on 
page 233. Write out a detailed criticism of the graphs or lack of 
them in it from the standpoint of their effectiveness with the public, 
showing just why they are good or liable to be unsuccessful. In the 
cases of the unsuccessful ones, or failure to use graphs where desirable, 
sketch graphs that would present the same data properly. 

REFERENCES FOR SUPPLEMENTARY READING 

Brinton, W. C. Graphic Methods for Presenting School Facts. Prac- 
tically all. 

Ellis, A. Caswell. ** The Money Value of Education." U. S. Bureau 
of Education Bulletin, 1917, No. 22. • 

King, W. I. Elements of Statistical Method, Chapter X. 

Rugg, H. 0. Statistical Methods Applied to Education, Chapter X. 



CHAPTER XII 

TRANSLATING STATISTICAL MATERIAL ON 
SCHOOLS FOR THE PUBLIC 

I. THE NEED OF TRANSLATION 

For the superintendent who wishes to present his school 
statistics effectively to the public, three devices are 
available. He may graph his material ; he may tabulate 
it ; he may translate it into words. Each procedure has 
its strengths and weaknesses. Each is best adapted to 
certain conditions. 

Translation is the most serviceable device in cases 
where it is difficult to secure or to have printed good 
tabulations or graphs. A translation of statistical 
material into words with a few mere numbers can be 
typewritten or set up in type anywhere with little effort. 
Such a translation can be read or spoken at any public 
meeting with no particular preparation. Furthermore, 
it can be so neatly expressed that persons hearing it can 
easily remember it and quickly pass it on to others, a 
thing not possible with tabulations and graphs. It can 
be so forcibly worded that it will arouse people to action. 

For example, the New York Survey devoted many 
pages to figures and graphs setting forth the results 
attained in arithmetic. But it is very doubtful if the 
whole or any part of this section could impress the 
average man as does the simple translation of the summary 

303 



304 School Statistics and Publicity 

by Mc Andrew : "It takes us less time to get a thing 
wrong here than it does in the average school system." ^ 

Statistical material on schools is often presented in 
words in such a way as to be clear to the trained school man 
and yet be unintelligible to the average man. If it is to 
be clear to the latter, it must be as definitely translated 
for him as must material from the Latin or other foreign 
languages, from a doctor's description in medical parlance 
of a disease, from fundamental political theories, or from 
scientific experiments in agriculture. We have various 
writers to translate the classics ; we have Woods Hutchin- 
son and other medical writers to translate medical 
knowledge ; we had Miinsterberg and James to translate 
psychology; we have various writers and speakers to 
translate political theories into the party campaign 
books; the agricultural colleges have numerous writers 
of bulletins to translate agricultural knowledge for the 
farmer and housewife. Have we not just as much need 
for translating school statistics into the language of the 
average man? 

Beyond clearness, the matter of force is very important. 
The translation must not only be clear to the man for 
whom it is intended, but it must take hold of him in some 
way. Force cannot be obtained by the mere repetition of 
tabulations in straight sentences of reading matter, how- 
ever much some state superintendents appear to think 
it can. This is not translation. The problem is really f 
the same problem as that of the life insurance compa- 
nies, the corporation seeking to influence the public, and 
advertisers in general, when they try to reach the public 
with statistical material. They do far more than merely 
express figures by words. 

1 The Public and Its Schools, p. 8 



Translating Statistical Material 305 

It cannot be too strongly stated that translation is 
not the same as definition or explanation. In defining 
statistical terms we simply aim to show precisely what 
we mean by them, often in terms just as technical. In 
explaining statistical terms we merely try to make our 
particular use of them clear to persons who already have 
the same definitions of them as ourselves, this too in 
language often just as technical. Good translation of a 
statistical term, of course, involves both definition and 
explanation, but it is more. It puts the emphasis on the 
fact that the meaning must be carried over into an entirely 
different language or set of expressions. For example, the 
median may be defined as the magnitude of the midpoint 
in a distribution. In any given frequency table or sur- 
face of frequency, it may be explained that the point 
marked " m " signifies the median. But the idea of the 
median in a set of superintendents' salaries may be 
translated for the average man by telling him that half 
the superintendents get more than that salary and half 
of them get less. 

The writer has never seen any discussion of this trans- 
lation phase of school statistics. Consequently, the 
treatment here is only a preliminary or tentative analysis, 
to be supplemented with illustrations from any source. 
For clearness and brevity, the points will be given rather 
dogmatically. 

II. SUGGESTIONS FOR GOOD TRANSLATIONS 

The main things to be kept in mind in working out 
good translations of school statistics are : 

1. The illustrations and images used must be of an 
elementary nature, or at least familiar to the people for 
whom the translation is being made. 



306 School Statistics and Publicity 

The announcement that retarded children in school are as thick as 
Ford automobiles or men wearing Masonic pins is intelligible any- 
where. On the other hand, the statement that retarded children are 
as thick as negroes in southern cities would be a very effective way of 
translating the facts to some southern audiences, but would hardly 
be of much value elsewhere. It would not even work in all southern 
cities, because the proportion of negroes in them varies from close to 
50 per cent down to less than 5 per cent. 

2. In some cases it may be necessary to use several 
illustrations in order to be sure of reaching all classes of 
people. 

3. Instead of representing a total by imagining an 
unreal extension of a familiar object, or by making up 
from familiar units an aggregate so large as to be in- 
comprehensible, it is usually better to employ some other 
unit. Often this other unit is one of time. 

It is of doubtful value to ask the average man to think of a line 
of school children eight hundred miles long ; of a schoolhouse as large 
as all the schoolhouses of the county put together; of a sheet of 
writing paper large enough to cover a township; of a lump of coal 
weighing as much as all the coal burned in one day in many schools ; 
or of a total of any sort reaching into the hundreds of thousands. 
Such translations are sometimes attempted. They might well be 
called Jack-and-the-beanstalk translations, for they are about as far- 
fetched. 

Practically all statistical totals needing to be translated will, on 
examination, be found to involve in some way units of length, area, 
volume, weight, and time. The aim should be to translate the total 
into another kind of unit that will keep it within the limits of com- 
prehension or experience of the ordinary man.^ Often a relatively 
larger unit of time that is forceful may be employed than in the case 
of the other kinds of units, because most people comprehend long 
stretches of time or the consumption of goods over long periods, 

1 For elaborating this point the writer is indebted to Mr. H. A. 
Webb, one of his students. 



Translating Statistical Material 307 

fairly well. The periods of time, however, must in general be well 
within the limits of the ordinary man's active life span. 

Consider the following translation of the average daily absence 
for the schools in Texas in 1911-13 : 

"Placed twelve feet apart, these white pupils absent every day 
from the schools of Texas would form a line extending across the 
state from El Paso to Texarkana, a distance of over eight hundred 
miles." 1 

The average Texan, even though he is from an early age accustomed 
to boasting of the size of the state, can have but a very hazy idea of 
the distance mentioned. Even if he has traveled from one city to 
the other, his idea is dependent chiefly upon his recollection of the 
time it took to make the trip, and a good part of this time he may have 
been asleep. In all likelihood, he has never seen children lined up 
twelve feet apart this way. It would probably be better for most people 
to consider these children as marching double file and say how many 
days it would take them to pass a given point. The average man has 
seen people marching double file in parades and has a good idea of 
about how fast they would pass. 

Instead of saying that, if the school costs of a certain city were 
represented by silver dollars lying side by side, they would extend 
the distance of ten blocks, it would be far better to say that at a cer- 
tain sum, say $50 a month, it would take a man a certain number of 
years to earn an equivalent amount of money. In this case, the 
average man has a much better idea of how long it takes him to earn 
money than he has of the distance dollars touching each other will 
extend. He has never seen silver dollars lined up along a street, but 
he has worked for wages and knows the value of his money. 

A very effective example of this sort is found in the Negro Year- 
Book for 1916-17? The compiler of this made a study of the average 
number of days that each negro child of school age attended school 
in each of the southern states. To make the smallness impressive, he 
calculated the number of years it would take the average negro child 
to complete the elementary school on the basis of eight grades and 
nine months to the school year, thus : 

1 White, E. v.: "A Study of Rural Schools in Texas," Bulletin 
of University of Texas, No. 364, Oct. 10, 1914, p. 20 

2 Page 233 



308 School Statistics and Publicity 

No. of yrs. it would take 
the average negro child to 
complete the elementary 
course in the public schools 
State provided for him 

Maryland 16 

Texas 18 

Virginia 18 

Georgia 19 

Florida 20 

North Carolina 20 

Alabama 22 

Louisiana 25 

South Carolina 33 

This could have been made more forcible by changing the figures to 
the age at which a negro entering school, when six years old, could 
complete the grades, as 22 for Maryland, 24 for Texas, etc. 

4. In cost statistics, it is sometimes advisable to 
minimize the total by expressing it in amount per small 
unit of time, usually a trivial sum. 

Thus daily papers and many weekly periodicals advertise "10 cents 
a week" and do not call attention to the total of $5.20 for the year. 

A church calls for "30 cents a week" and does not emphasize the 
fact that this reaches $15.60 for the year. 

The Y. M. C. A. announces that membership with all privileges 
costs "a nickel a day or the price of your daily cigar, " when the yearly 
total is from $15 to $18, without mentioning these latter figures. 

The Liberty Loan called for "a dollar a week" instead of stressing 
the $50 that would be laid aside for the year. 

The sum for the total is generally not stressed except in savings 
bank advertisements and such appeals, where the aim is to surprise 
the reader with the total saved, and not any amount that is expended. 

5. Absolute accuracy frequently has to be sacrificed 
to force and clearness in translation. 

Thus, if the percentage of negroes in a city population was 40, and 
the percentage of retarded children 35 or 45, the negro illustration 



Translating Statistical Material 309 

mentioned before would be accurate enough for translating the 
idea. 

The "nickel a day" of the Y. M. C. A. is adequate for any an- 
nual sum from $15 to $18. 

6. Practically all totals have to be translated through 
comparisons, using familiar objects or notions, before they 
can be understood or have much force for the average man. 

The statement that four per cent of all children of school age are 
mentally defective and need special attention would mean little to 
most men. But to say that every school system enrolling 500 chil- 
dren had among that number 20 defective pupils, or enough to equal 
the ordinary 6A grade taught by one teacher, would drive this fact 
home to the average man. 

The mere statement that there were on the average 64 children to 
each grade teacher in a school system would mean little to the people 
of the city and probably would receive no attention. However, if 
the statement were changed to read that there were enough surplus 
children over the standard number for the various classes to fill four 
classrooms, the fact would certainly impress citizens. 

The bare announcement of the amount spent for some item of 
school expense is not nearly so forceful as the statement that it is 
only a certain fraction of the ice cream and soda, liquor or tobacco 
expenditures of the town, if such figures or approximate estimates 
of them can be obtained from merchants. 

A southern county superintendent recently translated his valua- 
tion estimates as follows : 

"The total value of all school property in the county outside 
of the city is $34,420, which is $3080 less than half the value of 
the county jail and site, and $3000 less than one fourth of the 
value of the county courthouse, site, furniture and fixtures. 

" Of the $34,420 invested in school property in the county, only 
$15,665, or less than half, belongs to the state and county. In 
other words, the value of all the school property of those schools 
with titles vested in the state is $5335 less than the cost of three 
of her best motor trucks used in constructing good roads. 

"The total value of all supplies and equipment, including 
musical instruments and libraries, is $5875, which is $1125 less 



310 School Statistics and Publicity 

than the cost of one motor truck used in building the roads of 
the county. 

"All school equipment in the county outside the city of 

is equal in value to less than one seventh of the value 

of the machinery owned by the county and used in the making 
of good roads. 'Seven to one' is the ratio of the county's in- 
vestment in equipment for making roads as compared to her in- 
vestment in equipment for making men and women." 

The same superintendent translated the total area of all the school 
grounds in the county, 73 acres, by comparing it with the 180 acres 
of playgrounds in the property of two country clubs near the county- 
seat. 

In similar fashion, the increased enrollment of the public high schools 
of the United States in 1914 over 1913 would make a city as large as 
Chattanooga and Knoxville, Tennessee, combined. 

Dr. L. P. Ayres did a great service for the school survey movement 
when he used the following translation to show its results : 

"About seven years ago this retardation became one of the 
most widely studied problems of educational administration, and 
in the past four it has been one of the prominent parts of the 
school survey. During the entire period, hundreds of superin- 
tendents throughout the country have been readjusting the 
schools to better the conditions disclosed. 

"In these seven years the number of children graduating each 
year from the elementary schools of America has doubled. The 
number now is three quarters of a million greater annually than 
it was then. The only great organized industry in America that 
has increased the output of its finished product as rapidly as the 
public schools during the past seven years is the automobile 
industry." ^ 

The Warren County (Kentucky) Bulletin translates the losses 
through retardation as amounting to "thousands of years." This 
is probably more impressive than really comprehensible. 

Suppose a teacher is repeating every answer after every pupil, 
hour in and hour out, for days, as did some teachers observed by the 
author's students. To tell a citizen that such a teacher is wasting 

^ Ayres, I-.. P.: "A Survey of Surveys," Indiana University Bul- 
letin, Vol. 13, No. 11, p. 180 



Translating Statistical Material 311 

time would mean little. But to show him that if the teacher had 
forty pupils, she probably would have twenty of them reciting half 
a day each day; that if she repeated every answer, she could only 
cover about half as much as if she did not repeat ; that this meant 
virtually wasting a fourth of a day for every pupil ; that a fourth of a 
day for forty pupils amounted to ten days for one ; that this ten days 
was two weeks in school for one pupil ; that this teacher by her repe- 
titions was each day consuming or wasting the equivalent of one 
pupil's time for two whole weeks in school, — all this would mean a 
good deal to him. 

7. Many questions involving value, and particularly 
exhibits of loss or waste, can be profitably translated 
into a money equivalent. This is particularly true of all 
proposals involving an increase in school taxes, which 
must, of course, be addressed to the taxpayers. 

Thus, the need of health education and sanitation may be shown 
by translating the loss in money through death and sickness into the 
cost estimates furnished by Professor Irving Fisher of Yale, and 
others. This is well done in the Warren County (Kentucky) Survey, 
page 2. It was found there that in the year 1915 there were 155 
preventable deaths, and the potential loss occasioned by these deaths, 
according to Professor Fisher's estimate, was shown to be $263,500. 

The well-known chart of the United States Bureau of Education, 
which attempted to show that every day spent in school is worth 
$9 to a child, is a good example. The fallacies in it are likely to 
give actual pain to any one who knows much about statistical 
method, or who will use his common sense effectively. But it does 
translate the material into something that is intelligible and appealing 
to most people. It has been of great value for thousands of high 
school commencement addresses and campaigns for increasing school 
levies. 

In any propaganda for increasing school taxes, it is well to trans- 
late the increase into the respective amounts of money which will be 
due from men who already pay certain round sums for total taxes, 
the number in each class, thus : 

This increase in school taxes will require : 

50 cents more per year from each of the 1500 men who now pay a 
total yearly tax of $5 each. 



o 



12 School Statistics and Publicity 



$1 more per year from each of the 500 men who now pay a total 
yearly tax of $10 each. 

$2 more per year from each of the 250 men who now pay a total 
yearly tax of $20 each, etc., etc. 

III. EXAMPLES OF GOOD TRANSLATIONS OF SCHOOL 

STATISTICS 

In many of the school surveys which use modern statis- 
tical method, the technical statistical terms and results 
have not been translated so as to influence the average 
man. To just this extent they are certain to fall short 
of what a survey or review of a school claims to be. 
True, there are to be found in isolated places some very- 
successful translations of these terms But these are so 
scattered as not to be generally accessible. The remainder 
of this chapter aims to make a few of these available for 

general use. 

1. Sampling 

The following translation of the process of sampling 
was used by the author's students in preparing some 
material for a survey of one of the western cities : 

In giving the standard tests to the children of this city, it was found 
to be too great a task to test every child, as labor in grading the papers 
would be enormous. So certain schools and grades were chosen at 
random, and tests were given in these places. 

When a carload of wheat is being graded, the grader does not 
look at all the grains, as every one knows that would take too long. 
What he does is to take a few grains from each of several places well 
distributed throughout the whole lot of wheat, and to make his rating 
from these samples. This process has been found accurate enough so 
that it is constantly used in business without complaint from either 
the buyer or seller. The same thing is true in grading fruit. Not 
every apple or peach is actually looked at, or even every box, but 
only certain apples or peaches taken from certain boxes (determined 
at random), and the quality of these determines the grade assigned 
to the whole consignment. 



Translating Statistical Material 313 

These examples are well known to the people of this 
section, in all probability because of the amount of wheat 
and fruit raised in the state. After they have read such 
a translation, are they likely to doubt the validity of the 
sampling done in the school survey? 

Sampling may also be explained by comparing it with 
the process of taking a straw vote. Most men understand 
how this is done, how reliable it is, etc. 

2. The Average 

A familiar idea for translating the average is afforded 
by the " see-saw " or by the lever. Every one knows that 
the farther the person is from the object upon which the 
lever is resting, the more weight it takes on the other 
side to counterbalance him. That is, the center is what 
the physicist means by " center of gravity." It is the 
same with the average. The average is the balancing 
point of all the cases in a distribution, with their distances 
from it and their sizes taken into account. The farther 
an item is from the average, the more weight it has. 

Professor King uses the expression '' a type " for the 
average. 

3. The Median 

The following translations are taken from surveys: 

"Among the teachers in the elementary schools, the median or 
midway age is twenty-nine years, half of the teachers being twenty- 
nine years old or older, and the other half, twenty-nine years of age 
or younger." ^ 

"With teachers ranked in descending order according to size of 
salaries, the median salary is the salary received by the teacher half 
way down the line." ^ 

1 "The PubHc Schools of Springfield, Illinois," Springfield Survey, 
p. 59 2 "Financing the Schools," Cleveland Survey, p. 53 



314 School Statistics and Publicity 

The idea of a median was translated with the aid of a picture by 
Superintendent Womack in his Conway (Arkansas) Survey by taking 
his ungraded class and lining them up for height from low to high. 
The middle child was standing on a drain which ran out white in 
front, and there were two gaps in the line to indicate the quartiles. 
Any reader could look at the line of pupils and quickly get a clear 
notion of what was meant by median height. 

" The point above which and below which fifty per cent of the cases 
fall." 1 

** The median is the case which was found in the investigation to 
have as many cases below it as there are above it." ^ 

A very effective translation of the median can be se- 
cured by using a description of a typical person or school, 
which will have the median amount in each of a number 
of different qualities. The first example of this use of 
the median ever noted by the author was the description 
of the typical teacher in Professor Coffman's Social Com- 
position of the Teaching Population, pages 79-80. A por- 
tion of this description follows : 

The typical American male public school teacher ... is twenty- 
nine years of age, having begun teaching at almost twenty years of 
age, after he had received but three or four years of training beyond 
the elementary school. In the nine years elapsing between the age 
he began teaching and his present age, he has had seven years of ex- 
perience, and his salary at the present time is $489 a year. Both his 
parents were living when he entered and both spoke the English 
language. They had an annual income from their farm of $700, 
which they were compelled to use to support themselves and their 
four or five children. 

His first experience as a teacher was secured in the rural schools, 
where he remained for two years at a salary of $390 per year. He 

1 Cubberley, E. P.: "Survey of the Organization, Scope and Fi- 
nances of the Public School System of Oakland, California," Board of 
Education Bulletin, No. 8, June, 1915. 

2 Ellifif, J. D.: "A Study of the Rural Schools of Saline County, 
Missouri," University of Missouri Bulletin, Vol. 16, No. 22, p. 8, 
footnote 



Translating Statistical Material 315 

found it customary for rural teachers to have only three years of 
training beyond the elementary school, but in order for him to ad- 
vance to a town school position, he had to get an additional year of 
training. Etc., etc. 

This has been imitated since in many cases. Thus, in 
the American School Board Journal, for August, 1917, page 
70, there is a description of the typical Iowa high school 
principal, based on medians. The author has had this 
device used by many of his students in writing up in- 
vestigations. 

4. The Mode 

The mode may be translated as follows : A certain 
article of clothing is said to be in " fashion " when more 
people wear it than do without it. Likewise in a distribu- 
tion, the mode is the *' fashion " in cases ; more appear 
there than anywhere else. 

5. Spread or Dispersion or Variability 

There are wide differences in the wealth of the people 
of this country. Wealth of individuals varies all the way 
from Rockefeller and his millions to the poor street 
beggar. This variation in statistics is called the spread 
or dispersion. Now some agitators, if they had their 
way, would eliminate this spread by making all people's 
wealth equal. 

In the same way there are all sorts of variations in 
children in school work, for any particular line of work. 
It would be just as great an error as that of the agitators 
to reduce the variability in any one line so that all children 
were considered equal in performance. 

Professor J. F. Bobbitt, in the School Review for October, 
1915, page 508, uses the translation '' zone of safety " to 



316 School Statistics and Publicity 

indicate that on high school costs of instruction, it would 
be well to try to get within the middle 50 per cent. 
That is, he translates the spread between the quartiles by 
'' zone of safety." 

6. Correlation 

The Biblical phrase, " the first shall be last and the last 
shall be first," might be used to good advantage in trans- 
lating a perfect negative correlation. ^ 

The following is a good translation of a coefficient of 
correlation of .48 between abilities in shop practice and 
abilities in drawing: 

There is marked evidence that abilities in shop practice and drawing 
accompany each other. Students above the average in one group will 
tend to be above the average in the other. It is not known specifically 
in what way the two abilities are centrally connected, or to what ex- 
tent the presence of either one is an indication of the other .2 

EXERCISE 

Take the school report or school survey used in the exercises on 
pages 233 and 302. Write out a detailed criticism of the transla- 
tions of statistics or lack of them in it from the standpoint of their 
effectiveness with the public, showing just why they are good or liable 
to be unsuccessful. In the cases of the unsuccessful ones, or failure 
to use translations where desirable, make up translations that would 
present the same data properly. 

REFERENCES FOR SUPPLEMENTARY READING 

Ellis, A. Caswell. " The Money Value of Education." United States 

Bureau of Education Bulletin, 1917, No. 22. 
McAndrew, William. The Public and Its School. 

1 Suggested by one of the writer's students, Mr. L. A. Sharp 

2 Rusg, H. O. : Statistical Method Applied to Education, p. 257 



SELECTED AND ANNOTATED BIBLIOGRAPHY 

The aim in this is to give a minimum list of the simpler and more 
easily accessible materials. 



I. Statistical Method 

Chapman and Rush. The Scientific Measurement of Classroom 
Products. Silver, Burdett and Company, Boston, 1917. 

Contains excellent brief chapters on the theory of scales, their 
application in schools, and dangers incident to their use. Other 
chapters present the more important scales for measuring work 
in the formal subjects in the elementary school, describe processes 
for getting results, and show how the results may be used to 
better classroom Work. 

Elderton, W. p. and E.- M. Primer of Statistics. A. & C. Black, 
London, 1910. 

A brief, very simple, and readable treatment, with no special 
reference to education. 

King, W. I. The Elements of Statistical Method. The Macmillan 
Company, New York, 1915. 

An elementary, concise, straightforward treatment, but adapted 
more to economic or historical work than to school problems. 

Monroe, W. S. Educational Tests and Measurements. Riverside 
Press, Cambridge, Mass., 1917. 

A treatment which works in many of the elements of sta- 
tistics very simply and forcibly, under discussions of various tests 
and scales. 

RuGG, H. O. Statistical Methods Applied to Education. The River- 
side Press, Cambridge, Mass., 1917. 

An admirable book for its general purpose, emphasizing the 
problems of the school administrator. The statistical part proper, 
while written as much as possible in a non-technical style, is natu- 
rally carried to a much greater refinement and intricacy than are 

317 



318 School Statistics and Publicity 

necessary in preparing school statistics for publicity. However, 
the last chapter is very fine for publicity work. The bibli- 
ography covers all the main problems which superintendents need 
to study quantitatively, especially surveys, and is alone worth 
the price of the book. 
Thorndike, E. L. An Introduction to the Theory of Mental and 
Social Measurements. Teachers College, Columbia University, 
New York, 1913. 

A complete exposition of things needed in the fields indi- 
cated by its title. It has few direct applications for the admin- 
istrator, and is extremely difficult for the beginner in statistics. 

II. Calculating Tables 

Crelle, a. L. Rechentafeln. G. Reiner, Berlin, new edition, 1907. 

Gives products to 1000 by 1000. 
Peters, J. Neue Rechentafeln fiir Multiplikation und Division. 

G. Reiner, Berlin. 

III. Exercises and Problems 

Thorndike's Mental and Social Measurements and Rugg's Statistical 
Methods Applied to Education have problems in various places, 
some of which may be easily adapted for practice work. 

RUGG, H. O. Illustrative Problems in Educational Statistics. Pub- 
lished by the author. University of Chicago Press, 1917. 

IV. Graphic Methods 

Brinton, W. C. Graphic Methods of Presenting Facts. Engineering 
Magazine Company, New York, 1914. 

An excellent non-technical treatment, profusely illustrated. 
While not written especially for school men, its conclusions and 
suggestions are easily adapted to school problems. 
Ellis, A. Caswell. The Money Value of Education. Bulletin of the 
United States Bureau of Education, 1917, No. 22, Washington. 
Contains numerous charts used in educational campaigns. 
RuGG, H. O. Statistical Methods Applied to Education. The River- 
side Press, Cambridge, Mass., 1917. 

Chapter X presents numerous good examples. 



Selected Bibliography 319 

V. School Reports, General Treatments 

Bliss, D. C. Methods and Standards for Local School Surveys. D. C. 
Heath and Company, Boston, 1918. 

A simple but adequate treatment of the topics indicated, which 
will be of great value to the superintendent. Has many good 
tabulations and some graphs. 

Giles, J, T. A Statistical Study of School Reports from the Twenty- 
five Largest Cities of Indiana. Educational Administration and 
Supervision, Vol. II, pp. 305-311. 

Hanus, Paul H. School Efficiency. A Constructive Study. School 
Efficiency Series. World Book Company, Yonkers-on-Hudson, 
N.Y., 1913. 

A study of twenty-six widely selected city reports in the United 
States. 

Snedden, David S. and Allen, William H. School Reports and 
School Efficiency. The Macmillan Company, New York, 1908. 
A pioneer book in this field, now useful chiefly for its sug- 
gestions on what to include in a report and on using tabulations. 

VI. School Reports and Surveys Especially Valuable from a 

Publicity Standpoint 

The publisher is indicated for each item given here. For a copy 
of any other survey or report mentioned in the body of the text, 
address the superintendent of the school system concerned. 
Alabama. An Educational Survey of Three Counties in Alabama. 

Department of Education, Montgomery, Ala., 1914. 
Boston, Massachusetts. Report of a Study of Certain Phases of 

the Public School System of Boston, Massachusetts. Teachers 

College, Columbia University, New York City, 1916. 
Butte, Montana. Report of a Survey of the School System of Butte, 

Montana. By Strayer, G. D., and others. Board of School 

Trustees, 1914. 
Cleveland, Ohio. The Cleveland Education Survey. Ayres, L. P., 

Director. Published in twenty-five separate monographs by 

the Survey Committee of the Cleveland Foundation, Cleveland, 

Ohio. The following are especially valuable: 

Child Accounting in the Public Schools — Ayres 
Financing the Public Schools — Clark 



320 School Statistics and Publicity 



4 



Measuring the Work of the Public Schools — Judd 
The Cleveland School Survey (summary) — Ayres 

Dansville, New York. A Study — The Dansville High School. 
By Foster, J. M. F. A. Owen Publishing Co., Dansville, N. Y. 

Denver, Colorado. Report of the School Survey of School District 
Number One in the City and County of Denver. Part I, General 
Organization and Management; Part II, The Work of the Schools; 
Part III, The Industrial Survey; Part IV, The Business Manage- 
ment; Part V, The Building Situation and Medical Inspection. 
The School Survey Committee, Denver, Colorado, 1916. 

Des Moines, Iowa. Annual Report of the Des Moines Public Schools. 
For the year ending July 1, 1915. Board of Education, Des 
Moines, Iowa. 

Ellis, A. Caswell. The Money Value of Education. Bulletin of 
the United States Bureau of Education, 1917, No. 22. 

Grand Rapids, Michigan. School Survey. By a large staff. 1916. 

Janesville, Wisconsin. An Educational Survey. By Theisen, 
W. W., and staff of state department. Published by State De- 
partment of Public Instruction, Madison, Wisconsin. 

McAndrew, William. The Public and Its School. World Book 
Company, Yonkers-on-Hudson, N. Y., 1916. 

Minneapolis, Minnesota. Three Monographs on School Finance 
in Minneapolis. By Spaulding, F. E. Board of Education, 
Minneapolis, Minn. : 

A Million a Year 

Financing the Minneapolis Schools 

The Price of Progress 

Newburgh, New York. The Newburgh Survey. Department of 
Surveys and Exhibits, Russell Sage Foundation, 128 East 23d 
Street, New York City, 1913. 

Newton, Massachusetts. The Newton Public Schools. By Spauld- 
ing, F. E. 1912 and 1913, Newton, Mass. (Out of print.) 

New York City. Report of Committee on School Inquiry. By Hanus, 
Paul H., and others. School Efficiency Series, World Book 
Company, Yonkers-on-Hudson, N. Y. 

Ohio. Report of the Ohio State School Survey Commission. By Camp- 
bell, M. E., Allendorf, W. L., and Thatcher, C. J., 1914. 

Portland, Oregon. The Portland Survey. By Cubberley, E. P., 
and others, 1913. 



Selected Bibliography 321 

Report of the Committee on Uniform Records and Reports. U. S. 

Bureau of Education Bulletin, 1912, No. 3. 
RocKFORD, Illinois. A Review of the Rockford Public Schools, 1915- 

1916. Board of Education, Rockford, Illinois, 1916. 
Salt Lake City, Utah. Report of a Survey of the School System of 

Salt Lake City, Utah. By Cubberley, E. P., and others. Board 

of Education, Salt Lake City, 1915. 
San Antonio, Texas. The San Antonio Public School System. By 

Bobbitt, J. F. The San Antonio School Board, 1915. 
Springfield, Illinois. The Public Schools of Springfield, Illinois. 

By Ayres, L. P. Division of Education, Russell Sage Founda- 
tion, New York City, 1914. 
St. Louis, Missouri. Report of Survey of St. Louis School System. 

Board of Education, 1917. 
Texas. A Study of Rural Schools in Texas. By White, E. V. and 

Davis, E. E. University of Texas, Austin, Texas, 1914. 
United States. A Comparative Study of Public School Systems in 

the Forty-eight States. Division of Education, Russell Sage 

Foundation, New. York City, 1912. 

This is now out of print, but copies will doubtless be available 

for the superintendent at his state department of education or 

state university. Copies were sent to all the members of the 

state legislatures at the time of its issue, 1912. 



INDEX 



Note. — To save space, the words education, publicity, school, 
statistics, and teachers are in the main omitted from this index. In 
the numerous combinations in which they naturally occur, look for 
the next most significant word. 



Absence, translation for, 307 

Absurdities in rank-order combi- 
nations, 198 

Accuracy, 187-190, 230; errors 
in attempts at too great ac- 
curacy, 16-19 ; in graphs, 283 ; 
in translations, 308 

Advertising, school, error in bas- 
ing upon exceptional graduates, 
23 

Age-grade tables, emphasis in, 228 

Age-progress, table form for, 229 ; 
circle graphs for, 251 

Age, school, error in determining, 
5 

Ages of pupils, distribution table 
for, 120 ; graphs for, 121 

Aikins, Professor, 47 

Alabama, Survey of Three Counties, 
239, 250, 288, 289, 290 

Allen, W. H., 25, 27, 33, 72, 78, 
201, 202, 213, 216 

Alphabetical order in tables, 221- 
222 

Alternate columns in tables, 218 

Amarillo, Tex., 68 



American Book Company, graph, 
241 

Area graph for comparison on 
component parts, 245 ; for 
comparisons with circles, 251 

Arithmetic tests, 46, 47, 49 ; sur- 
face of frequency for, 117; 
frequency table for, 134; table 
for results, 232 ; variability in, 
171 

Arkadelphia, Ark., 8 

Artistic features in tables, 220 

Association of science and mathe- 
matics teachers, table to show 
growth, 232 

Atlanta, 300, 301 

Attendance, at teachers' associa- 
tions, error in computing, 14 ; 
school, blanks for showing, 72, 
73 ; error in indefinite units, 
4; perpetual graph device for 
showing, 298 ; problems of, 34 

Automobile graph, 290, 296 

Average, 141-147 ; advantages 
of, 145; computation of, long 
method, 141-142 ; computa- 



323 



324 



Index 



tion of, short method, 142 ; 
definition of, 141 ; disadvan- 
tages of, 146; errors in com- 
puting, 19-21 ; graphic repre- 
sentation of, 144; translation 
for, 313 

Average deviation, 156 

Averages, deviations from, 21-22 

Ayres, L. P., 5, 166, 241, 243, 
244, 251, 259, 291, 292, 310 

Ayres handwriting scale, 53 

Ayres spelling scale, 106 

Background for graphs, 241, 283 

Bagehot, Walter, 90 

Baltimore Survey, 248 

Bar graph, 237; cartoon effect, 
239, 260, 288; comparisons 
with, 243, 244, 253; com- 
ponent parts, use for, 239; 
curve effects with, 263 ; order 
of items for, 241 ; right and 
left form, 247, 248 

Bi-modal distribution, 147 

Bird's-eye view through tabula- 
tion, 209 

Birmingham, Ala., 210, 298 

Black, use of, on maps, 293 

Blanks, 71-78; making, 78; 
examples of good, 78; re- 
vision of, 42; vs. card index, 
79-81 

Bloomington, Ind., 172 

Bobbitt, J. F., 17, 22, 69, 70, 86, 
95, 96, 101, 162, 167, 315 

Bobbitt table, 18, 96, 103, 104, 
127, 130, 147, 152, 153, 156, 
161, 162, 165, 169, 223, 225, 
226, 240, 293, 295 



Bold-face type, use in tables, 219, 
227 

Boston, circulation of school re- 
port, 27 

Boston Report, 245, 246, 269 

Bowley, A. L., 189 

Bridgeport Survey, 229, 255, 257, 
258 

Brief, use of, in planning, 42 

Brinton, W. C, 116, 242, 272, 
276, 286; graphic methods of 
presentation, 242 ; rules for 
graphic presentation, 272-.275 

Buffalo, 213 

Buildings, lack of, cartoon effect 
for, 301 ; problems on, 34 ; 
table for, 213; units and 
scales for, 53 

Butte Survey, 117, 128, 130, 167, 
293 

By-products in collecting data, 
41 

Calculating devices, 191 
Calculating tables, 191 
Calculation, economies in, 190- 

192 
Card index vs. one blank, 79-81 
Carelessness in securing data, 15 
Cartoon effects in graphs, 256, 

287, 301 
Cartoons, economies in making, 

296 
Census, school, problems of, 34 
Central tendency, measures of, 

124-147 
Charts, time, 261 
Checking, in calculations, 190 ; 

on b!?.nks, 84 



Index 



325 



Chicago, University of, statistical 
method in School of Education, 
29 ; blank for rating teachers, 
270 

Cincinnati, 214 

Circle graph, 235, 259 ; for com- 
parisons, 250, 254, 255, 256, 
259; with cartoon effect, 258, 
259 

Cleveland, 5, 36, 174, 214 

Cleveland Survey, 118, 154, 
159, 251, 253, 256, 282, 294, 
313 

Coefficient of correlation, cal- 
culation of, 184, 185; ex- 
amples of, 183; meaning of, 
181, 182 

Coefficient of variability, 170 

Coffman, L. D., 314 

Collecting data with high school 
students, 86-88 

College degrees, error in com- 
parisons with, 8 

College salaries, error in getting 
average, 19 

Columbus Dispatch, 290 

Committee on standards for 
graphic presentation, 267 

Committee on uniform records 
and reports, 58 

Comparisons, simple, 240 (see 
also Relationships, 164-186) ; 
using component parts, 14, 266, 
267; using percentages, 13, 
14 ; using indefinite units, 
3-16; using relative position, 
269 ; using unsound treat- 
ment, 3-16; with bar graphs, 
239, 240, 243, 244, 253, 266, 



267, 269 ; with circle graphs, 
250, 256, 258; with cartoon 
effects, 256, 258; with curves, 
104, 262, 263, 266, 267; with 
triangle graphs, 257 

Component part graphs, 236, 
238, 239, 242, 245, 254, 255 

Composition scales, 46, 53 

Composition tests, frequency 
table for, 131 ; graph for, 168 

Concentric circle graph, 252 

Constant errors, 188 

Contests, judging, 11-12, 52, 
192 ff. 

Continuous series, 48 ; graph for, 
165 

Conway, Ark., Survey, 314 

Correlation, 173 (see also Co- 
efficient of Correlation) ; 
graphic devices for showing, 
175-181 ; Hke-signs, table for, 
178-179; translation for, 316 

Cost of instruction, 16, 17, 18, 
48, 55, 70, 96, 101, 104, 105, 
130, 150, 222, 223, 244, 307, 
308 

Cost of maintenance, 218, 264 

Cost per pupil, 263 

Cost records, cartoon effect to 
show value of, 287 

Costs, cartoon for, 290 ; errors 
in computing, 16; graphs for, 
288, 292; sampling for, 68; 
translation for, 307, 308 ; units 
and scales for, 55. 

Courtis arithmetic tests, 46, 47, 
142, 143, 149, 199; frequency 
table for, 134 ; graphs for, 172, 
266 



326 



Index 



Crayons for graphs, 296 

Crelle tables, use of, 191 

Cross-section paper, for computa- 
tion, 190 ; for large graphs, 296 

Cubberley, E. P., 263, 265, 314 

Current school reports, 200 

Curve effects with bar graphs, 
263, 266 

Curves, arbitrary signs for, 166; 
for comparisons, 262 ; for dis- 
crete scales, 104; standards 
for, 266-268 

Data, carelessness in securing, 
15 ; collection of, 33-89 ; econ- 
omies in collecting, 82-88 ; 
sources of, 58-61 

Dearborn, W. F., 106, 107, 108, 
180 

De Voss, J. C, 271 

Defectives, unanalyzed total for, 
3 ; number of, translation for, 
309 

Degrees, college and university, 
error in comparisons with 8 

Des Moines, 207, 210, 219, 257 

Detroit, 217 

Deviation, coefficient of, 170 ; 
measures of, 149-163 ; graphic 
representation of, 101, 137 ; 
measure for given distribution, 
162 ; translation for, 315 

Dimmitt, R. L., 298 

Discrete scale or series, definition, 
48 ; graphic representation of, 
103-105, 164 

Dispersion (see Deviation) 

Distance, map to show by time 
elements, 280 



Distribution, of cases on a map, 
277 ; of pupils, age-grade blank 
for, 73; of school moneys, 
error in, 5; of time in ele- 
mentary grades, graph for, 236 ; 
tables, 106-111 ; graphing. 111 

Dollar circle graphs for component 
parts, 254 

Dollar graph for proportionate 
parts with cartoon effects, 258, 
259 

Dollar proportionate parts table, 
230, 231 

Dollar-sign graph, 260, 289 

Dots for bar graphs, 292 

Double distribution table, 72 

Double entry table for receipts 
and payments, 211 

Economies, in collecting data, 82- 
88 ; in graphs, 296 

Education, value of, graph to 
show, 247 ; translation for, 311 

Educational investigations, knowl- 
edge needed for, 94 

Efficiency record of teachers, 
summarizing graph for, 720 

Elimination, bar graph for, 243; 
of high school students, table 
for, 217 ; problems of, 34 

Elliff, J. D., 314 

Emphasis in tables, 227 

Enrollment, errors in, 3-5; 
graphs, 248, 249, 252, 253, 261, 
294; map devices to show 
distribution, 277, 278 ; N. E. A. 
blank for, 74; translation for, 
310 ; units and scales for, 57 

Errors, constant, 188 ; variable, 187 



Index 



327 



Estimating reliability, 189 
Exaggerations in graphs, 283 
Expenditure graphs, 245, 259, 

265, 269 

Expenditures, average, 62 ; errors 
in comparisons, 4 ; problems 
of, 34 ; proportionate units 
and scales for, 54 ; sampling 
for, 68, 70; tables, 214, 215; 
translation for, 308 (see also 
Costs) 
Extreme range variation, 149 
Eyestrain, avoiding, 218, 227 

Fisher, Irving, 242 

Florida, 20 

Ford automobile translation, 306 

Fractions, errors in getting too 

small, 17 
Frederic, Wis., 88 
Frequency, surface of. 111, 116, 

123 ; multi-modal surface of, 

125; normal surface of, 117; 

skew surface of, 118-122 ; 

tables, 106-111 

Georgia, University of, 279 

Giles, J. T., 200 

Grading pupils, 10, 11, 21, 22, 

63, 66, 107, 108, 170, 196; 

blank for studying teachers' 

standards, 76 
Graphs, 234-301 ; check list for, 

274 ; economies in making, 

296; examples of good, 288; 

size of, 282 ; standards for, 

266, 272, 281; summarizing, 
268 

Gray, W. S., 48 



Grounds, size of, cartoon graph 

for, 260; translation for, 310 
Grouping, 41, 107-110, 210 
Gummed letters for graphs and 
charts, 297 

Haggerty, M. E., 171, 172, 176, 
199 

Hammond Survey, 220 

Handwriting scales, 45, 50, 53 

Handwriting tests, distribution 
table for, 128 ; surface of fre- 
quency at Cleveland, 118 

Hanus, Paul, 25, 27, 200, 201 

Harvard-Newton composition 
scale, 53 

Headings, for graphs, 281 ; for 
tables, 209, 219; form for 
printing, 216 

Health education, lack of, trans- 
lation for, 311 

Heating, 22 

Herrick, W., 26 

High school enrollment, graph 
for, 293 

High school training, value of, 
graph to show, 247 

Hillegas composition scale, 53, 
131 

Histogram or column diagram, 
111-116, 122 ; check form, 114, 
115 

Illiteracy, cartoon graph to show, 
288 ; error in treatment of, 13, 
16 

India ink for graphs and charts, 
297 

Indiana, 199, 200 



328 



Index 



International Harvester Company 
graph, 247 

Jingle fallacy, 47 
Judging contests, 11, 12, 52, 
192 ff. 

Kelley, F. J., 184, 271 

Key numbers, for blanks, 75; 

for tables, 219 
King, W. I., 36, 38, 39, 41, 91, 

92, 95, 97, 145, 313 
Kirk, Jno. R., 65 

Length of school year, bar graph 
for, 291 ; translation for, 307 

Lettering on graphs, 296 

Library Bureau, 59 

Library statistics, error in in- 
definite units, 10 

Lighting space, graphs for, 286, 
295 

Lines, clear in graphs, 297; 
dividing, in tables, 218, 219 

Louisville, 6, 231, 244, 258, 283 

Lowry, F. C, 289 

MacAndrew, Wm., 300, 304 

Maclin, E. S., 300 

Maintenance cost, per cent of, 

table for, 218 
Maps, 275 ff. ; use of black on, 

293 
Marking pupils, 10, 11, 21, 22, 63, 

66, 107, 108, 170, 196; blank 

for studying teachers' standards 

on, 76 
Masonic pin translation, 306 
Maxwell, Wm., 25, 26, 37 



Median, advantages of, 138; 
computation of, 129-137; def- 
inition of, 128; disadvantages 
of, 139 ; graphic representa- 
tion of, 137 ; translation of, 313 

Median deviation, 155 

Medical inspection, unanalyzed 
totals in, 3 ; problems in, 35 

Membership, table for showing 
growth in, 232 

Memphis, 166, 265 

Meyer, Max, 11, 63 

Minneapolis, 63, 70, 145 

Minnesota, 9 

Missouri, University of, grading 
system, 11 ; State Department 
of Education, 279 

Mobile, 27 

Mode, advantages of, 126; def- 
inition, 124 ; computation of, 
125; disadvantages of, 127; 
graphic representation of, 125 ; 
translation of, 315 

Monroe, W. S., 136, 271 

Monument graph, 249 

Multimodal surface of frequency, 
-125 

Nashville, Tenn., 77, 119-121, 

280 
National Education Association, 

58, 74, 78 
Neatness in tables, 220 
Negroes, schooling for, 15, 307 
Newburgh Survey, 259, 260, 261 
Newton, Mass., 40, 102, 103, 

261, 263, 288, 300 
New York Survey, 27, 303 
Normal schools, 2,7, 150 



Index 



329 



Normal surface of frequency, 117- 
119 

Oakland Survey, 218 

Ohio Survey, 259 

Old Man Ohio cartoon, 290 

Omitting important factors, 15 

Order of items for bar graph, 241 ; 

for tables, 221, 223 
Over-age, error in method of 

determining, 5 

Paper letters for charts and 

graphs, 297 
Peabody (George) College for 

Teachers, 277 
Pearson coefficient of correlation, 

158, 185 
P. E., or probable error, 155 
Penmanship (see Handwriting) 
Percentage tables, graphing, 167 
Percentages, errors in, 13-15 
Percentile deviations, 153 
Percentiles, 153 

Perpetual attendance graph de- 
vice, 298 
Phelps, S. J., 86 
Phi Beta Kappa, 11 
Planning statistical treatment, 38- 

43 
Playgrounds, cartoon graph for, 

260; size of, translation for, 

310 ; units and scales for, 53 
Population of school district, 

units and scales for, 56 
Portland Survey, 51, 52, 224, 227, 

230, 282 
Presentation of school statistics 

to public, errors in, 24 



Probability surface of frequency, 

118 
Problem, statistical, how to state, 

39 
Property, value of, translation 

for, 309 ; value of, graph for, 289 
Proportionate parts table, 231 
Public indifference, 26-27 
Puckett, W. F., 5 

Q or Quartile Deviation, 151, 153 ; 
graphic representation, 137 ; 
translation for, 315 

Quartiles, 151; graphic repre- 
sentation, 137 

Questionnaire method, value of, 
59 ; blank for, 65 ; sampling 
for, 66 

Range, measures of, 149-163 
Rank-order combinations of data, 

192-198 
Rating of teachers, summarizing 

graph for, 270 
Receipts and payments, problems 

of, 34 ; table, 211 
Records, 58, 287 
Red Cross, 279 
Relationships, 164-186 
Relative position, bar graph for, 

269 
Rehabihty, 187-190 
Reports, 25-27, 40 
Reproducing graphs for the public, 

297 
Retardation, error in determining, 

6; graph for, 166, 237, 257; 

problems of, 35; translation 

for, 306, 310 



330 



Index 



Revenues, errors in comparison, 
15 

Rice, Dr., 9, 45 

Rockford Review, 231, 236, 252, 
264, 279, 285 

Rubber stamps for graphs and 
charts, 297 

Rugg, H. 0., 67, 79, 182, 316 

Ruled blank book, 77 

Ruler strip device for use on re- 
ports, 85, 86 . 

Rural school work, map for, 279 

Russell Sage Foundation, 166, 
243,244,257, 259, 260, 261 

Salaries, janitors', 257 ; principals', 
257; teachers', 22, 109, 110, 
125, 150, 154, 290; teachers', 
error in indefinite units, 6, 7 ; 
teachers', units and scales for, 
54 

Salt Lake City Survey, 122, 125, 
168, 219, 232, 237, 282, 293 

Sampling, 22-24, 62-71 

San Antonio Survey, 17, 22, 86, 
167, 225 

San Francisco Survey, 66 

Scales, 24, 43-53, 100; discrete 
and continuous, 48; examples 
of, 53-57; graphic represen- 
tation of, 101, 285; objective, 
44 ; subjective, 44 

Schedules, teachers', error in com- 
puting with indefinite units, 7 

Schooling, value of, translation 
for, 311 

School problems, 34-37 

School statistics, errors in, 1-24 ; 
errors in presentation of, 24 ; 



need for better, 1-32 ; value 

of, for superintendents, 30-31 
Scoring data, economies in, 82-84 
Semi-inter-quartile range, 151 
Sequence in tables, 221 
Seymour, F. 0., 68 
Sharp, L. A., 316 
Shaw-Walker Com.pany, 59 
(T = Greek sigma = abbreviation 

for Standard Deviation 
Signs, arbitrary, for graphs, 

299 
Skew distribution, 148 ; devia- 
tions for, 159 
Skewness, 118-122 
Skew surface of frequency, 118- 

122 
Smoothed surface of frequency, 

119 
Smoothing graphs, 113 
Snedden, David, 25, 27, 33, 72, 

78, 201, 202, 213, 216 
South Bend Survey, 209, 221 
Spaulding, F. E., 40, 43, 63, 70, 

71, 103, 145 
Special classes, 35 
Spelling, errors in lack of units 

for, 9 ; scale, 106 ; tests, 45, 

122, 167, 169 
Spread, measures of, 149-163; 

translation for, 315 
Springfield Survey, 211, 243, 257, 

313 
Standard Deviation, 158 
Standard tests, sampling in, 66 
Standards, errors in striving for, 

17-19 
Statistical method, reliability of 

results with, 187, 189, 190; 



Index 



331 



value of, and when to use, 19- 
24, 28, 29, 30, 33, 36, 37, 38- 
40, 91-93, 95, 97-99, 100, 200 

Step method, economy for com- 
putation, 191 

Step on a scale, meaning of, 24, 49 

Stone tests, 117 

Strayer, G. D., 62, 79, 119 

Street maintenance, table for, 
225 

Students, use of, for statistical 
work, 192, 299 

Subjective scales, 46 

Summarizing data on blanks, 75, 
76 

Summary tables, 221 

Summer session enrollment, error 
in indefinite units, 3 

Supervision, problems of, 35 

Surface of frequency, 111-122 

Symmetrical distribution, 117 

Tables, distribution, 106; of 

frequency, 106 ; series of, 220 
Tabulation, 209, 233; for the 

public, 203, 204, 206-233 
Taxes, increase in, bar graph for, 

261 ; increase in, translation 

for, 311 
Tax rate, errors in comparison 

with, 7, 8 ; graphs for, 102, 

265 ; table for, 224 ; units and 

scales for, 56, 57 
Teachers College, use of statistical 

method at, 28 
Teaching staff, units and scales 

for, 54 
Technical methods needed in 

school statistics, 19-24, 90-99 



Tests, rank-order combinations 

for, 192 ; summarizing graphs 

for, 168, 271 
Textbooks, graph for cost of, 242 
Thermometer graph, 102, 288 
Thorndike, E. L., 30, 45, 46, 47, 

48, 49, 79, 92, 109, 144, 189, 

191, 206 
Thorndike handwriting scale, 45, 

53 
Tie rankings, 195 
Title of graph or chart (see 

Headings) 
Time charts, 261 
Time spent on each subject, units 

and scales for, 56 
Time unit for translations, 306 
Totals, forms for emphasizing, 

216 ; unanalyzed, error in, 2-3 
Training of teachers, graphs for, 

256, 259 ; units and scales for, 

54 
Translation of statistics for the 

pubHc, 303-316 
Triangle graphs, 254 
Truancy, problems of, 35 
Two-way tables, 72 
Type, measures of, 124-147 

Unclassified items table, 214, 215 
Unequal things, errors in con- 
sidering equal, 9-10, 47 
Uniform records and reports, 233 
Units, errors in, 3-12, 16; ex- 
amples of, 53-57; how to de- 
termine, 43-57 
Updegraff, Harlan, 70 
U. S. Bureau of Education, 59, 
120, 247, 248, 311 



332 



Index 



1 



Uselessness of statistics in school 
reports, 200 

Valedictorian, determining, 192 
Valid scale, 47-48 
Vanderbilt University, 279 
Variability, of children, graph 

for, 293; coefficient of, 170; 

translation for, 315 
Variable errors, 187 
Variation, measures of, 149-163 
Variations, errors in neglecting, 

21-22 ; in size of type for 

tables, 232 
Variety in graphs, 286 
Virginia, 9, 286 

Waste, in school statistics, 24- 
26; in teaching, translation 
for, 311 

Wax crayons for graphs and 
charts, 296 



Wealth, units and scales for, 56 ; 

real and assessed behind each 

$1 spent on schools, table for, 

51,52; real, table for, 223 
Webb, H. A., 306 
Weighing machine cartoon, 289 
Weighting factors in rank-order 

combinations, 197 
Wisconsin, Experiment Station, 

287; State Department of 

Public Instruction, 88 
Withdrawals, error in determining 

age, 6 
Womack, J. P., 314 

Y. M. C. A., 280, 308, 309 

Zero line in graphs, 263, 283, 

284 
Zero point on scale, 46, 47 
Zeros, use in tables, 216, 217 
Zone of safety, 18, 315 



5477 



LIBRARY OF CONGRESS 



021 334 61 



67 



